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+{"text":"\\section{Introduction}\nPerhaps the most remarkable demonstrations of coherent interaction between atoms and photons are electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA) and coherent population trapping  \\cite{boller1991observation,fulton1995effects,alezama1999eia,renzoni1997coherent}. These processes can be interpreted as a consequence of quantum interference -- they are based on the fact that an optical field can transform atomic states such that an atomic transition can be entirely suppressed and subsequent absorption eliminated. Quantum interference shows an exceptional sensitivity to frequency shifts, including those induced by magnetic fields. This makes atomic ensembles excellent tool for magnetometry, with potential application across research fields as diverse as biomedicine, seismology, defense, and general metrology \\cite{kitching2011atomic,wiesendanger2011single}. \n\nAtomic magnetometers can now reach excellent sensitivites, comparable to, and even surpassing those of superconducting quantum interference devices (SQUIDs) \\cite{drung1990low,pannetier2004femtotesla,kominis2003subfemtotesla}.  Since the first demonstration of EIT-based scalar magnetometers \\cite{fleischhauer1994quantum}, various schemes have been reported, relying on the zero field resonance observed in Hanle-type experiments \\cite{alipieva2003narrow,gateva2007shape}, on optical pumping \\cite{acosta2006nonlinear,afach2015highly,bison2018sensitive,zhang2019multi}, or the nonlinear Faraday effect in a manifold of a single ground state \\cite{novikova2001compensation,pustelny2006pump,budker1998nonlinear,budker2000sensitive,novikova2000ac,pustelny2008magnetometry}. Miniaturisation presents a challenge for atomic magnetometers, but over the last two decades devices have been developed that combine extreme sensitivity with minute detection volumes \\cite{kominis2003subfemtotesla,shah2007subpicotesla}.\n\nMost atomic magnetometers perform scalar magnetic field metrology, i.e.~determined the magnetic field along a pre-defined axis. Simultaneously measuring the strength and direction of a magnetic field would be of great importance in specific areas such as satellite navigation and biological magnetic field measurement \\cite{budker2007optical,le2013optical}.\nThe direction of a magnetic field can be addressed by vector magnetometers, first demonstrated in \\cite{lee1998sensitive}. Since then, various schemes characterized by EIT or its counterpart, EIA, have been extensively studied \\cite{dimitrijevic2008role,yudin2010vector,cox2011measurements,margalit2013degenerate}. The full vector nature of a magnetic field may be accessed by simultaneously probing the magnetic field in orthogonal directions by separate probe beams.  Alternatively, adding an external transverse magnetic field (TMF) can make EIT-based methods sensitive to different magnetic field components by considering polarization rotation or resonance amplitudes \\cite{yudin2010vector,cox2011measurements}. \n\nIn this work we explore the possibility to detect both the strength and alignment of a magnetic vector field from the interaction of a warm atomic vapor with a vector beam (VB), i.e.~a light field that has a polarisation pattern that is varying across the beam profile.    \nThe interaction of vector beams with atoms is a relatively new concept \\cite{zhan2009cylindrical,wang2020vectorial}, which has been used to explore spatial anisotropy \\cite{fatemi2011cylindrical,wang2018optically,yang2019manipulating,wang2019directly,wang2020optically}, nonlinear effects \\cite{shi2015magnetic,stern2016controlling,bouchard2016polarization,hu2019nonlinear} and quantum storage \\cite{parigi2015storage,ye2019experimental}. \n\nOf particular interest to this work is the extension of EIT, conventionally observed as spectral features with homogeneously polarised probe beams, to spatially resolved EIT resulting from inhomogeneously polarised VBs.  This effect has been observed both in cold \\cite{radwell2015spatially} and warm \\cite{yang2019observing} atomic systems. In the former case, a weak TMF closes the EIT transitions, thereby generating phase-dependent dark states and, in turn, spatially dependent transparency. As the spatially observed transparency patterns and applied magnetic fields are directly coupled, this offers the possibility of detecting magnetic fields from absorption profiles  \\cite{clark2016sculpting,castellucci2021atomic}.\n\n\nIn this paper, an experimental setup is presented to visually observe the magnetic field based on Hanle resonances in a warm atomic vapor. Importantly, we analyze spatially resolved absorption patterns instead of the time-resolved spectrum, which is fundamentally different from other aforementioned methods.\nBy employing VBs, we show that the absorption pattern is sensitive to the TMF strength, visible particularly in the degree of absorption, whereas maximal transmission remains unchanged. It's worth noting that the above results will change depending on experimental parameters, \\textit{e.g.} increasing the temperature of the gas should lead to a reduction of transparency throughout the whole beam profile. Furthermore, the transmitted pattern of VBs can be rotated arbitrarily according to the alignment of TMF. For the general case in the current work, the spatial magnetic field can be decomposed into a TMF and a longitudinal magnetic field (LMF) according to the quantization axis. The absorption patterns and corresponding polar plots can then be analyzed to recover the full magnetic field information. Such a procedure could prove to be a powerful tool to measure the three-dimensional (3D) magnetic distribution, and can even be applied in room temperature atomic vapors, simplifying future atomic magnetometer design.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figure1.eps}\n\\caption{The schematic of the experimental setup and atomic energy levels. M: mirror; HWP: half-wave plate; QWP: quarter-wave plate; L: lens; PBS: polarization beam splitter; PD: photodetector; SMF: single mode fiber; VRP: vortex retarder plate; CCD: charge-coupled device camera; MFS: Magnetic field shielding; PM: Projection measurement. SAS: Saturated absorption spectroscopy; VBG: Vector beam generation.}\n\\label{fig1}\n\\end{figure} \n\n\\section{Experimental setup}\nThe experimental setup is shown in Fig. \\ref{fig1}. The output of a frequency locked 795 nm external cavity diode laser is sent through a single-mode fiber (SMF) to improve the mode quality of the Gaussian beam. \nAfter the fiber, the beam passes through a half-wave plate and a polarizing beam splitter (PBS) to adjust the beam intensity and fix the polarized state of the beam as horizontal polarization. A telescope is applied to expand the beam size and the achieved high-quality Gaussian beam waist is 4 mm. The VBs are generated by sending the linearly polarized beam through a vortex retarder plate (VRP), a liquid-crystal-based retardation wave plate with an inhomogeneous optical axis which displays an azimuthal topological charge \\cite{marrucci2006optical,marrucci2006pancharatnam}. \nThe laser frequency is locked to the $5{S_{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2}}},F = 2 \\to 5{P_{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2}}},F' = 1$ transition of the $^{87}$Rb D1 line.\nThe Rb cell has a length of 50 mm. A three-layer $\\mu$-metal magnetic shield is used to isolate the atoms from the environmental magnetic fields. The temperature of the cell is set at 60\u00b0C with a temperature controller. A solenoid coil (not shown in figure) inside the inner layer generates a uniform LMF, oriented along the light propagation direction, $\\mathbf{k}$. A TMF, in the plane perpendicular to $\\mathbf{k}$ and coverin the whole cell, is generated by two pairs of orthogonal Helmholtz coils, each pair independently controlled by a high precision current supply driver.  By adjusting the current ratio, and hence the horizontal and vertical TMF component, it is possible to produce an arbitrary TMF. The power of the incident laser beam is 3.4 $\\rm mW\/cm^{2}$ ( $\\approx$ 0.75 $I_{\\rm sat}$). After passing through the cell, the spatial intensity distribution of the beam is recorded by a charge-coupled device camera (CCD). \n\nThe polarization distribution of the probe VBs can be reconstructed by measuring the Stokes parameters, which represent the full polarization information of  the light \\cite{milione2011higher}. Experimentally, the Stokes parameters can be obtained by using the projection measurement system consisting of a QWP, a polarizer and a CCD. Fig. \\ref{fig2} (a) shows the polarization distribution of the generated VB with $m=$1, which is also known as a radially polarized beam. Here, $m$ is the polarization topological charge of the VB. It can be seen that this distribution varies periodically with the azimuthal angle in the plane of the beam. The electric field vector of the VBs considered here can be expressed as:\n\\begin{equation}\n\\mathbf{E}(r, \\phi,z)={E_0}(r, \\phi,z)\n\\begin{pmatrix}\n \\cos(m\\phi)\\\\\n \\sin(m\\phi)\\\\\n 0\n\\end{pmatrix}.\n\\label{1}\n\\end{equation}\nHere $r$ is the radial distance, $\\phi$ denotes the azimuthal angle and $z$ is the propagation distance. The position-dependent complex amplitude of the light is ${E_0}(r, \\phi,z)$, where $m$ is an integer.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure2.eps}\n\\caption{The experimental results of the radially polarized beam in presence of TMF. (a) Intensity and polarization distribution without atoms. (b) - (h) Intensity distributions after passing through atoms under vertical TMF of varying strength: ${\\rm B}_{\\rm TMF}$ = 0 mG, 23 mG, 61 mG, 123 mG, 146 mG, 206 mG and 230 mG, respectively. (i) The dependence of transmitted intensity for two selected regions against ${\\rm B}_{\\rm TMF}$.}\n\\label{fig2}\n\\end{figure}\n\n\\section{Experimental results}\n\nNow we turn to how the magnetic field influences the interaction between the vector beam and the atoms: the transmission in particular. Firstly, when the magnetic field is not applied, there is very little absorption of the vector beam, as compared to the profile without atoms (Fig. \\ref{fig2} (a) and (b)). As a vertical TMF is applied to the atoms however, a petal-like transmission pattern gradually appears, as shown in Fig. \\ref{fig2} (c)-(h). In general, one predicts $2\\times m$ petals, with the exception of \n$m=0$. In our case, we consider $m=1$, and so we observe a two-fold symmetry, as considered in more detail below.\n\nIncreasing the magnitude of the TMF, we observe that maximum transmission always occurs in the region where the linear polarization is parallel or antiparallel to the TMF. The strongest absorption occurs in regions where the local linear polarization is perpendicular to the TMF axis, and it increases with increasing magnitude of the TMF. We note that positive and negative TMFs both lead to the same pattern, as atomic transitions respond to the alignment but not orientation of linear polarization. The variation of transparency  and absorption  are plotted against the TMF strength (from $-230$ mG to $+230$ mG) in Fig. \\ref{fig2} (i). To make a systematic comparison, a point of maximum transmission and absorption respectively is chosen, and the corresponding probe intensity ($\\rm{I_{\\parallel}}$ and $\\rm{I_{\\perp}}$) is determined, averaged over a square area of 25 pixels, to reduce experimental error. The red curve then shows the local sensitivity of the VBs to the TMF strength, in agreement with the Hanle-EIT profile. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure3.eps}\n\\caption{The experimental results of the radially polarized beam in presence of an LMF. (a) - (f): the intensity distributions after passing through the atom vapor under the varied LMF: ${\\rm B}_{\\rm LMF}$ = 0 mG, 50 mG, 100 mG, 120 mG, 160 mG and 200 mG, respectively. (g) The dependence of transmitted intensity for whole beam against the ${\\rm B}_{\\rm LMF}$.}\n\\label{fig3}\n\\end{figure}\n\nAs expected, the absorption of the optical ${\\mathbf E}$-field components aligned with the $\\mathbf B$-field orientation is independent of the magnitude of the applied magnetic field strength. We can therefore monitor these spatial positions to consider the effects of an LMF. Accordingly, applying an LMF of varying magnitude, between $0~mG$ and \n$200~mG$, we observe uniform absorption across the whole beam, that increases with the magnitude of the TMF (Fig.~\\ref{fig3}). The variation of transparency for the whole beam is shown in Fig. \\ref{fig3} (g), displaying the same Hanle-EIT profile as Fig. \\ref{fig2} (i). Such results are adequately described by the Zeeman effect and are no different to prior experiments that rely on linearly polarized light.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure4.eps}\n\\caption{Transmission profiles as function of TMFs alignment. (a) and (b): intensity and polarization distributions for vertical and diagonal TMF alignment. (c) Image axis of the transmission profile as a function of TMF alignment. Insets: examples of observed transmission profiles.}\n\\label{fig4}\n\\end{figure}\n\nThe transmitted vector beam, as a whole, is highly sensitive to the TMF however, particularly in regard to the $B$-field's alignment. To characterize the axis of the TMF, we select a radially polarized VB which generates the two-petal pattern after passing through the vapor cell. We then define the {\\it image axis} as the line that passes through the maximal transmission regions of the two petals and the beam center. As mentioned previously, the visibility of the transmission profile can be controlled by the magnitude of the TMF, with a stronger magnetic field corresponding to stronger maximal absorption. We set the TMF to $230~mG$ to ensure maximal contrast, allowing us to identify the image axis as precisely as possible. We further note that the linear polarization of the transmitted region is also parallel to the image axis, providing an alternative way to identify the TMF axis. \n\nFig. \\ref{fig4} (a) and (b) show the relationship between the TMF alignment and the transmitted polarization, as reconstructed from CCD measurements of the Stokes parameters. Fig. \\ref{fig4} (c) shows the angle of the image axis (green arrow) when rotating the TMF axis from $0$ to $\\pi~$rad. Moreover, the VBs' polarization can also be manipulated by rotating the half wave plate and the VRP, producing the same rotational results as when the angle of the TMF axis is fixed. The axis of TMF can thus be easily observed and the TMF's strength can be measured similarly to the procedure outlined in Fig. \\ref{fig2}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure5.eps}\n\\caption{The experimental results of the radially polarized beam in presence of the spatial magnetic field with fixed strength ($\\left | \\mathbf{B} \\right |=230mG$). (a): $\\left | \\mathbf{B} \\right |=0mG$. (b) - (f): transmitted patterns with $\\theta = \\pi\/6, \\pi\/4, \\pi\/3, 5\\pi\/12, \\pi\/2$, respectively. (g) Polar plots for patterns at different angle $\\theta$ at the radius indicated in (f).}\n\\label{fig5}\n\\end{figure}\n\nTo visualise arbitrary magnetic field alignments, we can combine our observations for a TMF and an LMF with further experiments that consider an arbitrary inclination angle, $\\theta$, between the $\\mathbf{B}$-field and the propagation axis, $\\mathbf{k}$. For a TMF along the vertical axis, $\\theta$ then denotes the angle in the $y-z$ plane and the magnetic field can be written as $\\mathbf{B}=\\mathbf{|B|} (0, \\sin{\\theta}, \\cos{\\theta})^T$, as reported in Fig. \\ref{fig5}. \nIn the absence of a magnetic field, there is no petal-like pattern and the transmission of the radially polarized VB is uniformly distributed along the azimuthal angle, similarly to Fig. \\ref{fig2} (b) and Fig. \\ref{fig3} (a). \n\n\nWe then set the strength of the magnetic field to $\\mathbf{|B|}$ = 230 mG. When $\\theta = 0$, the magnetic field is aligned to $\\mathbf{k}$ , corresponding to a pure LMF, destroying Hanle resonances and resulting in strong homogeneous absorption as discussed in the context of Fig. \\ref{fig3}. By increasing $\\theta$, the petal-like pattern gradually appears according to the strength of the TMF component. In the case of $\\theta = \\pi\/2$, the magnetic field is purely transverse, and the transmission profile is the same as for Fig. \\ref{fig4}. As expected, regions where the polarization is perpendicular to the $\\mathbf{B}$-$\\mathbf{k}$ plane experience maximal absorption. However, with larger angle $\\theta$, the TMF component of $\\mathbf{B}$ increases and induces transparency in regions with parallel polarization. The relationship and sensitivity of transmitted patterns to the angle $\\theta$ are captured in Fig. \\ref{fig5} (g). The visibility of the pattern also depends on the magnitude of the field. A rotation of the magnetic field around the azimuthal angle $\\phi$ would result in a corresponding shift of the absorption pattern. Visual inspection of the absorption pattern therefore gives maximal information on the magnetic field alignment, subject to the symmetry of the probe pattern.\n\nSo far, we have considered only radially polarized VBs, but similar observations hold for general VBs. We demonstrate this in Fig. \\ref{fig6} by comparing VBs with differing topological charge: $m$ = 1 and $m$ = 2.  \nFig. \\ref{fig6} (a) and (d) show the polarization and intensity profile of the generated VBs with without atoms, respectively. These donut-like profiles change very little without appropriate shielding from magnetic fields, as shown in Fig. \\ref{fig6} (b) and (e).  However, in the presence of a TMF, the transmitted beams show a two-fold and four-fold petal pattern: as shown in Fig. \\ref{fig6} (c) and (f). Polar plots of the absorption profile with atoms, for $m$ = 1 and $m$ = 2, are shown in Fig. \\ref{fig6} (g) and are in agreement with the original observation in cold atoms \\cite{radwell2015spatially}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure6.eps}\n\\caption{Polarization and intensity profiles for VBs with different polarization topological charges with  $m$ = 1 (top row) and $m$ = 2 (bottom row). (a) and (d) profiles of VBs without atoms, (b) and (e) after passing through atoms in the absence of a magnetic field, (c) and (f) petal-like patterns under $\\mathbf{B}_{\\rm TMF}$ = 230 mG. (g) Polar plots of the absorption profile in (c) along the radius of largest contrast.}\n\\label{fig6}\n\\end{figure}\n\n\\section{Theoretical interpretation and discussion}\n\nAs is now well known, atom-light interaction is strongly polarization dependent. There are however, infinitely many ways to decompose a field's polarization and to choose an atom's quantization axis. For the former, the spherical-basis has many useful properties \\cite{yudin2010vector,lee1998sensitive,auzinsh2010optically}.  Here, light polarized perpendicularly to the atom's quantization axis drives a superposition of $\\sigma^{+}$ and $\\sigma^{-}$ transitions ($\\Delta m_{\\rm F} = \\pm1$), while light polarized parallel to the quantization axis drives the associated $\\pi$ transition ($\\Delta m_{\\rm F} = 0$). In the absence of a magnetic field, it is convenient to choose the quantization axis, $z$, along the propagation axis, $\\mathbf{k}$, so that any light, $\\mathbf{E}$,  polarized in the $x-y$ plane is simply formed from the superposition of two orthogonal circular components with equal amplitude and a varying phase difference. In the presence of a magnetic field however, it is helpful to choose the quantization axis along the axis of the magnetic field, $\\mathbf{B}$,  so that the interaction is only dependent on the angle between $\\mathbf{B}$ and $\\mathbf{E}$. \n\n\nThus, we define the optical field in the spherical basis$\\left \\{ \\mathbf{e_{0}=e_{z}, e_{\\pm1} = \\mp (e_{x}\\pm \\textrm{i} e_{y})\/\\sqrt{2}} \\right \\}$ \\cite{yudin2010vector}:\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E} \n&=E_{0}\\left((\\cos{\\alpha})\\mathbf{e_{0}}+\\frac{\\sin{\\alpha}}{\\sqrt{2}}(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\\right),\n\\label{2}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{e}_{i}~\\forall i \\in \\{x,y,z\\}$, are the basis unit vectors in Cartesian coordinates and $\\mathbf{e}_{j}~\\forall j \\in \\{0,+1,-1\\}$ represent the spherical basis with the quantization axis set by the magnetic field. Here, $\\pi$-polarized, left and right circularly polarized light corresponds to $\\mathbf{e}_{0,+1,-1}$ respectively, $E_{0}$ is the amplitude of light, $\\alpha$ is the angle between $\\mathbf{B}$ and $\\mathbf{E}$ and $\\beta_{1}$ and $\\beta_{2}$ are the phase of each circular polarization. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{figure7.eps}\n\\caption{Excitation scheme for the LMF and the TMF. Coherent dark state (b) and decoherent state (c) induced by Zeeman splitting. Bare dark state without (e) and with (f) Zeeman splitting. In presence of the magnetic field,\nmagnetic sublevels are shifted by an amount $\\mu_{\\rm B}g_{\\rm F}m_{\\rm F}B$, where $\\mu_{\\rm B}$ is the Bohr magneton, $g_{\\rm F}$ is the Land$\\acute{e}$-factor, and $B$ is the magnetic field strength.}\n\\label{fig7}\n\\end{figure}\n\n\\subsection{Interaction under an LMF}\n\nAs shown in Fig. \\ref{fig7} (a), when an LMF is applied, we choose the quantization axis  along the LMF, coinciding with the propagation direction of the light. In this case, all the linearly polarized components of the probe VBs are perpendicular to the quantization axis, and the light will connect all Zeeman sublevels of the $F = 2 \\to F^{'} = 1$ transition via multiple $\\Lambda$ schemes with simultaneous $\\sigma_{\\pm1}$ excitations \\cite{cox2011measurements,dancheva2000coherent,meshulam2007transfer}. According to the light polarization in Eq. (\\ref{2}), the VBs shown in Eq. (\\ref{1}) can be rewritten as:\n\\begin{equation}\n\\mathbf{E} = \\frac{E_{0}}{\\sqrt{2}} (\\cos{(m\\phi)} + \\sin{(m\\phi)})(-{\\rm e}^{-i \\beta_{1}(\\phi)}\\mathbf{e_{+1}} + {\\rm e}^{+i \\beta_{2}(\\phi)}\\mathbf{e_{-1}}),\n\\label{3}\n\\end{equation}\nwhere, $\\beta_{1}$ and $\\beta_{2}$ are dependent on the azimuth.\n\nWhen the LMF is zero, the Zeeman sublevels are degenerate, and atoms are pumped into a non-absorbing state induced by coherent population trapping. This is similar to the case of standard EIT, where the left and right circularly polarized components of an optical field resonate with the atomic levels to form the $\\varLambda$ structure shown in Fig. \\ref{fig7} (b). Here, a dark state due to coherent superposition of atomic energy levels is formed, which causes transparency of the whole VBs' profile. Increasing the strength of the LMF results in splitting of the Zeeman sublevels and an effective detuning as shown in Fig. \\ref{fig7} (c). Thus, the coherent dark state is destroyed, the atoms can now interact and this leads to absorption of the probe VBs. For any linear polarization exciting the $\\sigma_{\\pm}$ transitions, the transparency is sensitive to the magnitude of the magnetic field and shows the Hanle-EIT profile \\cite{anupriya2010hanle}.\n\n\\subsection{Interaction under a TMF}\n\nWhen a pure TMF is applied, as shown in Fig. \\ref{fig7} (d), the quantization axis is chosen to be aligned with the axis of the TMF. In contrast to the former case, the interaction of the linearly polarized components of the probe VBs is strongly dependent on the azimuthal angle. The components whose linear polarization is parallel to the TMF axis operate on the $\\pi$ transition \\cite{happer1972optical,huss2006polarization,yin2016tunable}, while the orthogonal components activate the $\\sigma_{\\pm}$ transitions. Other linearly polarized components can be considered as superpositions of these special cases and the inclined angle between the $\\mathbf{B}_{\\rm TMF}$ and the $\\mathbf{E}$ determines which transition is dominant. Here, the situation of perpendicular components is similar to interaction under the $\\mathbf{B}_{\\rm TMF}$ and shows the same sensitivity to the magnetic field strength. Assuming the $\\mathbf{B}_{\\rm TMF}$ is along the $y$ axis and combining Eq. (\\ref{1}) and Eq. (\\ref{2}), then the $\\mathbf{E}$-field of the VBs can be rewritten as:\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}&={E_{0}}\\sin{(m\\phi)}\\mathbf{e_{0}}+\\frac{E_{0}}{\\sqrt{2}}\\cos{(m\\phi)}(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\\\\\n&=E_{0}\\sin{(m\\phi)}\\mathbf{e_{\\parallel}}+E_{0}\\cos{(m\\phi)}\\mathbf{e_{\\perp}},\n\\label{4}\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{e_{\\parallel}}=\\mathbf{e_{0}}$ and $\\mathbf{e_{\\perp}}=(-{\\rm e}^{-i \\beta_{1}}\\mathbf{e_{+1}}+{\\rm e}^{+i \\beta_{2}}\\mathbf{e_{-1}})\/\\sqrt{2}$.\nThe first term in Eq. (\\ref{4}) is the linear component with polarization direction along the TMF axis driving $\\pi$ transitions. In this situation, atoms are optically pumped into the stretch states by means of spontaneous emission and removed from the optical transition. Here, another type of dark state, due to strong optical pumping, is formed which will cause transparency of parallel components and shows insensitivity to the magnetic field (Fig. \\ref{fig7} (e) and (f)). The second term in Eq. (\\ref{4}) is the linear component with polarization perpendicular to the TMF axis. In this position, the analysis is similar to the interaction under the $\\mathbf{B}_{\\rm TMF}$ as discussed before. Increasing $\\mathbf{B}_{\\rm TMF}$ enhances the absorption (destroying the coherence) of perpendicular components and the splitting of the beam profile, since parallel components are always transmitted. Thus, the transmission profile of the probe VBs under the non-zero TMF fellows the equation as:\n\\begin{equation}\nI\\propto\\left |\\sin{(m\\phi)}\\right |^{2},\n\\label{5}\n\\end{equation}\nsatisfying the $2m$ sinusoidal transmission profile observed in Fig. \\ref{fig6} (g).\n\n\\subsection{Interaction with arbitrarily oriented B-fields and associated applications}\n\nGenerally, when $\\mathbf{B}$ is not applied along the axis of light propagation, it can be decomposed into contributions of an LMF and a TMF. Thus, for the general case, the interaction of the light with atoms can be viewed as a combination of the cases discussed in sections $\\mathbf{A}$ and  $\\mathbf{B}$, including all three transitions. A coherent dark state only appears when $\\mathbf{B}=0$ and forms from pure $\\sigma_{+}$ and $\\sigma_{-}$ transitions. Frequency detunings (Zeeman splitting) and $\\pi$ transitions induced by the magnetic field break the coherence of these dark states. Although the $\\pi$ transition associated with the TMF induces bare dark states which are insensitive to the magnetic field.\n\nBased on the above analysis, the transmitted pattern of VBs after passing through the atoms is strongly dependent on the $\\mathbf{B}$, making this configuration a potentially useful tool for exploring spatially varying magnetic fields. The radially polarized beam whose polarization distribution resembles a compass would be used as the probe and the transmitted pattern has two petals whose orientation clearly show the axis of the TMF. Besides, by comparing maximum intensity of the transmitted pattern with initial intensity of the beam in the same position, the angle between the magnetic field and the plane of the beam profile can be easily obtained as shown in Fig. \\ref{fig5} (g). By combining the included angle with the axis of TMF, the axis of the spatial $\\mathbf{B}$ could be defined. Not only the alignment of the magnetic field, but also its strength influence the observed transmission pattern. The strength of the magnetic field  is associated with the intensity of the region perpendicular to two petals, since the polarization of this part is always perpendicular to the quantization axis set by the $\\mathbf{B}$. The transparency of this region depends on the Zeeman splitting of atoms influenced by the magnetic field, which means the magnitude of the magnetic field can be deduced by measuring the intensity of this region.\n\n\nUltimately, the alignmnt and strength of a spatial $\\mathbf{B}$-field can be seen and quantified from the transmitted vector beam. There are experimental limitations however. First of all, the measurement range of the magnetic field is limited by the atomic coherence. In this experiment, the Zeeman energy levels were used to build the atomic coherence, but hyperfine levels could expand the range of measurement of this configuration. Secondly, the direction of the spatial $\\mathbf{B}$-field can not be obtained here. Since only resonant light is used in this experiment, the frequency detuning induced by spatial $\\mathbf{B}$ has the same influence on both $\\sigma_{+}$ and $\\sigma_{-}$ transitions. After passing through atoms, the linear polarization can always be obtained which indicates two circular components have equal amplitudes and experience same absorption. Further study will be carried out to solve this problem by utilizing detuned light and measuring the polarization ellipse of transmitted parts \\cite{selyem2019three, clark2016sculpting}.\n\n\n\\section{Conclusion}\n\nIn conclusion, we have investigated the transmission properties and pattern formation of vector beams in an atomic vapor, as influenced by a magnetic field. In particular, there are two limiting cases: corresponding to two kinds of dark state. When an LMF is applied, an incoming probe beam undergoes uniform absorption, and the coherence between Zeeman sublevels (coherent dark states) can be destroyed by increasing the strength of the LMF. Applying a TMF however, will generate bare dark states and produce a petal-like pattern which is dependent on the azimuthal angle and topological charge of the polarization. The general case, where the magnetic field is applied along an arbitrary axis, is also studied: revealing the general influence of a spatial $\\mathbf{B}$-field on transmitted patterns. Thus, the information about both the alignment and the strength of a spatial $\\mathbf{B}$-field can be seen in the transmitted pattern of a vector beam, providing a powerful tool in the investigation of 3D magnetic fields. Recent works on chip-scale VBs generation \\cite{chen2020vector} and atomic components \\cite{garrido2019compact,stern2019chip,mcgilligan2020laser} would be an exciting next step for realizing miniaturization. We also note that similar effects are seen in related work in the diamond (nitrogen-vacancy center) \\cite{chen2020calibration} and cold atoms \\cite{castellucci2021atomic} carried out, confirming the suitability of VBs for visual observation of the alignment of magnetic fields in 3D space.\n\n\\section*{Funding} This work was supported by the National Natural Science Foundation of China (92050103, 11774286, 11534008, 11604257 and 11574247) and the Fundamental Research Funds for the Central Universities of China. FC and SF-A acknowledge financial support\nfrom the European Training Network ColOpt, which is funded by the European Union (EU) Horizon 2020 program\nunder the Marie Sklodowska-Curie Action, Grant Agreement No. 721465.\nTWC acknowledges support by the National Research, Development and Innovation Office of Hungary (NKFIH) within the Quantum Technology National Excellence\nProgram (Project No. 2017-1.2.1-NKP-2017-00001). \n\n\\section*{Disclosures} The authors declare no conflicts of interest.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace*{-0.5pt}\n\\noindent\n\tIn spite of its great successes, the Glashow-Weinberg-Salam (GWS) model, based on the \ngroup $SU(2) \\otimes U(1)$, seems somewhat unsatisfactory, in that its lagrangian contains a number of \narbitrary parameters and several terms that seem, at least at first sight, rather \\letra{ad hoc}. There have \nbeen different types of ideas to improve this situation. The most popular one by far has been to \nuse a large group of which $SU(2) \\otimes U(1)$\\ is just a small subgroup. Less popular has been the \nintroduction of symmetries that do not enlarge the group too much, a step-at-a-time policy, so to \nspeak. This has been done along three different lines, depending on what additional symmetry is \nimposed, be it the group $SU(2)_L \\otimes SU(2)_R$,\\citar{1} a discrete symmetry that restricts the model's \nparameters and allows for the calculation of mass corrections to quarks,\\citar{2} or \\su3. Here we \nshall concentrate our efforts on this last idea, not following the usual Yang-Mills scheme,\\citar{3} but \ninstead considering a modification of it.\n\n\tIt is interesting that \\su3\\ contains $SU(2) \\otimes U(1)$\\ as a subgroup and that it naturally \nassigns the correct hypercharge $Y$ and isospin $T_3$ quantum numbers to the electron and the \nneutrino if we place them in its fundamental representation so that they form the chiral triplet\n\\ecuacion{\n\\psi_L = \\pmatrix{ \\nu_L \\cr e_L \\cr e_L^c }\n\\label{chiral}\n}.\n\n\tIt is necessary to express the $e_R$\\ degree of freedom using the antiparticle formulation $e_L^c$\\ in order \nto obtain the hypercharge with the correct sign, as we shall soon explain. The main problem that \npresents itself immediately when one tries to represent the GWS model using \\su3\\ stems from \nthe boson sector, because 8 vector fields are required to gauge the theory, one for every generator \nof the adjoint representation. This means that there would be four more boson vector fields than \nare experimentally observed.\n\n\tA different approach is based not on Lie but rather on graded groups\\citar{4} and uses $SU(2\/1)$. \nThe excessive number of vector bosons introduced by the gauging of \\su3\\ is reduced by \nassuming that some of them are scalar instead of vectorial. It is possible to use the Higgs bosons \nas gauge fields, thus adding to the logical simplicity of the model. Fermions are placed in the \nfundamental representation of $SU(2\/1)$\\ using the nonchiral triplet\n\\ecuacion{\n\\psi = \\pmatrix{ \\nu_L \\cr e_L \\cr e_R }\n\\label{nonchiral}\n}, \nthat differs from the chiral one in the third component. The generators of the fundamental \nrepresentations of \\su3\\ and $SU(2\/1)$\\ differ in only one component of one generator. We are \ngoing to use the generators $T^a=\\frac{1}{2} \\ld{a}$, where the $\\ld{a}$ are the usual Gell-Mann matrices, so that \nthey are normalized according to\n\\ecuacion{\n\\widetilde{\\mathop {\\rm Tr}\\nolimits}\\, T^a T^b = \\frac{1}{2} \\delta_{ab}\n\\label{norma}\n},\nwhere we have put a tilde over the trace symbol to distinguish it from the traces over Dirac \nmatrices that we shall begin to use next section. The two generators that differ are both diagonal: \nthe last \\su3\\ Gell-Mann generator\n\\ecuacion{\n\\ld{8} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (1,1,-2)\n\\label{gene8}\n},\nand its partner in $SU(2\/1)$\n\\ecuacion{\n\\ld{0} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (1,1,2)\n\\label{gene0}\n}.\n\n\tThe isospins of $\\nu_L$, $e_L$\\ and $e_R$\\ are $\\frac{1}{2}$, $-\\frac{1}{2}$ and $0$, and the hypercharges are $1$, $1$ and $2$, respectively, \nprecisely as given by $\\ld{0}$. The electric charges of these particles can be calculated from the Gell-Mann-Nishijima \nrelation $Q=T_3-\\frac{1}{2} Y$, that relates the charge $Q$ to the isospin $T_3$ and the \nhypercharge $Y$. From these considerations it is clear that $\\ld{8}$ would assign a hypercharge $-2$ for \n$e_R$, that has the wrong sign, which is the reason why we had to use the antiparticle formulation \n$e_L^c$\\ in \\refeq{chiral}. The $1\/\\sqrt{3}$ normalization factor neatly goes into converting the single coupling constant \n$g$ of the model into the GWS model's two coupling constants $g$ and $g'\\approx g\/\\sqrt{3}$.\n\t\n\tFrom the beginning it was noticed the graded group approach presented two serious \ndifficulties: The first one was the so-called ``sign problem'', that arises from the fact that in non-abelian \ngauge (graded) theories the vector boson kinetic energy is constructed from the trace \n(supertrace) of the product of generators, that are (are not) positive-definite. A matrix \nnormalization that is not positive-definite will give the wrong sign to the kinetic energy of some of \nthe vector bosons.\\citar{5} The second difficulty, that we are going to call the ``statistics problem'', had \nits origin in the fact that, if one is trying to reproduce the GWS model, Higgs fields have to be \nplaced in the odd (or a-type) sector\\citar{6} of the adjoint of $SU(2\/1)$, and therefore necessarily have to \nanticommute amongst themselves. On the other hand, they have spin zero and must therefore \nobey Bose-Einstein statistics, or problems with unitarity of the scattering matrix will ensue; thus, \nin contradiction with the graded group requirement, they have to commute among themselves. To \novercome these difficulties was the motivation for much effort at the time.\\citar{7} Many different ideas \nwere tried such as adding extra odd dimensions to the spacetime manifold, and taking the \nGrassmann fields to be ghosts and not bosons, but, since neither difficulty could be resolved \nwithout causing worse difficulties, eventually interest on the subject waned.\n \n\tA few years ago a new branch of mathematics, noncommutative geometry,\\citar{8} was \ndeveloped. Its methods have been applied to the direct product of spacetime and spaces with a \ndiscrete number of points, with the result that some sectors of the standard model\\citar{9} have been \nobtained. It has also been applied to other areas of theoretical physics.\\citar{10} In the former case \nthere appears a graded algebra of forms invariant under the algebra $su(2\/1)$. The GWS model \nwith its $SU(2) \\otimes U(1)$\\ local gauge invariance appears if we define the lagrangian to be the trace \n(\\letra{not} the supertrace) of group invariants. This way the sign problem is solved \\letra{ab initio}. It is not \nclear to us just how successful this approach is in simplifying or unifying the Standard Model. \nAlgebraic structures and ideas vary from paper to paper, sometimes even by the same author, so \nthat, at this stage, we hesitate upon taking a position. \n\n\tAn attempt at generalizing a Yang-Mills theory using a graded gauge group will \ninevitably lead to the sign and the Higgs' statistics problems. Thus the question of attempting a \ngeneralization of the Yang-Mills covariant derivative using a non-graded Lie group arises \nnaturally.\\citar{11} A Higgs field assigned to the adjoint of the Lie group will now have the correct \nstatistics since it would be even (that is, c-type). Furthermore, since Lie groups are invariant \nunder traces, not supertraces, the kinetic energies of all the bosons will come out with the right \nsign. In this paper we show how to construct generalized Yang-Mills theories that are invariant \nunder local gauge transformations of a Lie (not graded) group and use a covariant derivative with \nboth scalar and vector fields. In the construction of these theories we also honor the condition, \nwhich we consider to be of an essential nature, that the lagrangians do not contain any \ndifferentiation operators acting indefinitely to the right. Terms of this type arise from powers of \nthe covariant derivative, but we require that somehow they all cancel out. It goes without saying \nthat such theories are also required to be Lorentz-invariant. It turns out that the only way to have \na covariant derivative contain both vector and boson gauge fields is for it to transform as a \n4-vector contracted with the Dirac matrices, that is, as a slashed 4-vector.\n\n\tWe apply these ideas to try to unify and reduce the number of different kinds of \nterms of the GWS model. If now one uses \\su3\\ as a gauge group, one \\letra{almost} obtains the GWS \nmodel, the difference being that the hypercharge of the Higgs boson comes out wrong. An even \nmore interesting result is that, if one uses the group \\uu3\\ and chooses a certain representation of \nthe generators, the GWS comes out exactly, plus an additional scalar boson that is totally \nuncoupled both from all the other particles of the model. This unified theory has only two terms: \nthe fermion kinetic energy, that is, the covariant derivative between two fermion fields, and the \nboson kinetic energy, that is, traces of powers of the covariant derivative, what is often called the \ncurvature. We have not added a new level of symmetry breaking and in this sense we\nhave not performed a ``unification'' in the traditional sense: this theory has the\nsame number of Higgs bosons as the original GWS and they break the symmetry in\nbasically the same way. However, with the new covariant derivative the GWS can be\nwritten in a simpler way in a larger group.\n\nIn this paper we do not go into the quark sector of the standard model. The presence\nof two right quarks (as opposed to only one right lepton) per family results in a\ndifferent, more complicated situation, that we plan to address in a future. \n\n\tIn section 2 we show how to write a traditional abelian gauge theory with the \ncovariant derivative transforming as the tensor product of two spinorial transformations. In \nsection 3 we introduce the concept of scalar fields as gauge fields. In section 4 we present non-abelian \nYang-Mills theories with mixed gauge fields. In section 5 we present two attempts at \nunification of the GWS model in the context of these ideas, using the Lie groups \\su3 and \\uu3. \nFinally, in section 6 we make a summary and conclude with a couple of remarks of what is wanting \nwith the model.\n\n\n\\section{A vector field transforming under the tensor product of two spinorial representations}\n\\noindent\n\tIn this section we are going to rewrite the quantum electrodynamics lagrangian\n\\ecuacion{\n{\\cal L}_{QED} = \\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi - \\frac{1}{4} F^{\\mu \\nu} F_{\\mu \\nu}\n\\label{lagqed}\n},\nwhere $D_\\mu \\equiv \\deriv{\\mu} + ie A_\\mu$ and $F_{\\mu \\nu} \\equiv \\deriv{\\mu} A_\\nu - \\deriv{\\nu} A_\\mu = -ie^{-1} [D_\\mu, D_\\nu]$ using only contractions of 4-vectors \nwith Dirac matrices, that is, avoiding dot products between the 4-vectors themselves. The reason \nfor this is that there does not seem to be a way of generalizing gauge theories keeping the \ncovariant derivative in the vector representation of the Lorentz group. One has to learn how to do \neverything with spinorial representations.\n\n\tThe fermion field $\\psi$ transforms under a local \\uu1\\ Lie group, so that, if $U=e^{-i\\alpha(x)}$ \nis an element of this group, then $\\psi \\to U \\psi$. To maintain gauge invariance it is necessary that the \nvector potential undergoes a gauge transformation, too, of the sort\n\\ecuacion{\nA_\\mu \\to A_\\mu + e^{-1} \\deriv{\\mu} \\alpha\n\\label{transfgauge}\n},\n\n\tFrom this transformation law and the definition of $D_\\mu$, it is evident that the covariant derivative \nmust transform as\n\\ecuacion{\nD_\\mu \\to U D_\\mu U^{-1}\n\\label{Dgauge}\n}.\n\n\tWe call a differentiation operator that acts only on immediately succeeding \nfunctions, but whose action then stops and does not differentiate any further functions to the \nright, \\letra{a restrained operator}. Likewise, we call a differentiation operator \\letra{unrestrained} if it keeps \nacting indefinitely to the right. As an example of the latter take the $\\deriv{\\mu}$'s in the covariant \nderivatives in \\refeq{Dgauge}, that are acting to the right for we do not know how far. It is not admissible to \nleave unrestrained operators in a lagrangian because, first, what they can mean physically or \nmathematically is not clear, and, second, they are not gauge invariant. The boson kinetic energy in \n\\refeq{lagqed} is proportional to $[D_\\mu,D_\\nu][D^\\mu,D^\\nu]$, each of whose factors is gauge invariant. On the other \nhand $D_\\mu D_\\nu$ is not gauge invariant as can be seen by direct calculation. The cause for this different \nbehavior is evident from the equation\n\\ecuacion{\n[D_\\mu,D_\\nu] f = \\deriv{\\mu} A_\\nu f - A_\\nu \\deriv{\\mu} f - \\deriv{\\nu} A_\\mu f + A_\\mu \\deriv{\\nu} f = (\\deriv{\\mu} A_\\nu) f - (\\deriv{\\nu} A_\\mu) f\n\\label{DmuDnu}\n},\nwhere it is seen how four unrestrained operators result in two restrained ones, thanks to Leibnitz' \nrule.\n\n\tIf we represent by $S$ an element of the Lorentz spinor transformation group, so \nthat $\\psi \\to S \\psi$ under a Lorentz transformation, then, due to the homomorphism that exists between \nthe vector and spinor representations of the Lorentz group, we have that $\\hbox{$\\not \\! \\! A$} \\to S \\hbox{$\\not \\! \\! A$} S^{-1}$. This \nhomomorphism allowed Dirac to write the equation of motion of electrons. But, is it possible to \nwrite the \\letra{boson} kinetic energy with the vector potential transforming this same way? The \nfollowing theorem answers this question in the affirmative:\n\n\\vspace*{12pt}\n\\noindent\n{\\bf Theorem:} Let $D_\\mu = \\deriv{\\mu}+B_\\mu$, where $B_\\mu$ is a vector field. Then:\n\\ecuacion{\n(\\deriv{\\mu} B_\\nu - \\deriv{\\nu} B_\\mu) (\\partial^\\mu B^\\nu - \\partial^\\nu B^\\mu) = \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 \\hbox{$\\not \\! \\! D$}^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^4\n\\label{Teo}\n},\nwhere the trace is to be taken over the Dirac matrices. Notice the partials on the left of the \nequation are restrained, the ones on the right are not. To prove the Theorem we expand the \ncovariant derivatives on the right side of \\refeq{DmuDnu}, and then use the following trick, which makes the \nalgebra manageable, in this and in more difficult examples in other sections. First, we define the \ndifferential operator $O \\equiv \\partial^2+2B \\cdot \\partial + B^2$. Notice that it does not contain any contractions with \nDirac matrices, so that $\\mathop {\\rm Tr}\\nolimits O = 4O$, $\\mathop {\\rm Tr}\\nolimits O (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$}) = 4O(\\partial \\cdot B)$, etc. It is not difficult to see then that \n$\\hbox{$\\not \\! \\! D$}^2 = O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})$, where the slashed partial is acting \\letra{only} on the succeeding slashed field. Now:\n\\ecuarreglo{\n\\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 [O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})] - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits [O + (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})]^2 &=& 2(\\partial \\cdot B)^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits [(\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})(\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! B$})] \\nonumber \\\\\n&=& (\\deriv{\\mu} B_\\nu - \\deriv{\\nu} B_\\mu) (\\partial^\\mu B^\\nu - \\partial^\\nu B^\\mu)\n\\label{prueba}\n}{}\nand we have finished proving the Theorem. \n\n\tWith the aid of the Theorem we can rewrite the QED lagrangian in the form\n\\ecuacion{\n{\\cal L}_{QED} = \\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi + e^{-2} \\left( \\frac{1}{32} \\mathop {\\rm Tr}\\nolimits^2 \\hbox{$\\not \\! \\! D$}^2 - \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^4 \\right)\n\\label{lagQED}\n},\nwhose Lorentz invariance can be proven using $\\g{\\mu} \\to S \\g{\\mu} S^{-1}$ and the cyclic properties \nof the trace. We prove, as an example, the Lorentz invariance of $\\hbox{$\\not \\! \\! D$}^2$: $\\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^2 \\to \\mathop {\\rm Tr}\\nolimits S \\hbox{$\\not \\! \\! D$} S^{-1} S \\hbox{$\\not \\! \\! D$} S^{-1} = \\mathop {\\rm Tr}\\nolimits \\hbox{$\\not \\! \\! D$}^2$. \nGenerally speaking, one could say that the cause for the term with \nthe squared trace in \\refeq{Teo} is the requirement that all differentiation operators be restrained.\n\n\n\\section{Abelian gauge theory using scalar boson fields}\n\\noindent\n\tWe want to construct an analogue to quantum electrodynamics, but using scalar \nbosons as gauge fields. This is, of course, not the so-called scalar electrodynamics, where the \nfermion matter fields are substituted by scalar bosons. Here it is the gauge field itself that has \nbecome a real scalar boson $\\varphi$. The transformation group is the same as in last section, and again \nto maintain gauge invariance \\refeq{Dgauge} must hold. We now define the covariant derivative to be\n\\ecuacion{\nD_\\varphi = \\hbox{$\\not \\! \\partial$} - e \\g5 \\varphi\n},\nand we require the gauge field to transform as $\\g5 \\varphi \\to \\g5 \\varphi - ie^{-1} \\hbox{$\\not \\! \\partial$} \\alpha$. These conditions immediately \nassure us that transformation \\refeq{Dgauge} holds in this case, too. There is an interesting point to be made \nhere: if we now substitute this covariant derivative in the QED lagrangian \\refeq{lagQED}, that was designed \nfor \\letra{vector} fields, we obtain the usual lagrangian for the interaction between a fermion and a real \n\\letra{scalar} boson field. That is, if we \\letra{assume} the lagrangian of the interaction to be\n\\ecuacion{\n{\\cal L}_{S} = \\overline{\\psi} i D_\\varphi \\psi + e^{-2} \\left( \\frac{1}{32} \\mathop {\\rm Tr}\\nolimits^2 D_\\varphi^2 - \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits D_\\varphi^4 \\right)\n\\label{lagS}\n},\nthen the expansion of the covariant derivative results in \n\\ecuacion{\n{\\cal L}_{S} = \\overline{\\psi} i \\hbox{$\\not \\! \\partial$} \\psi - e \\overline{\\psi} i \\g5 \\varphi \\psi + \\frac{1}{2} (\\deriv{\\mu} \\varphi)(\\partial^\\mu \\varphi)\n},\nafter a bit of algebra. This calculation is similar to the one done last section, but with the $\\g5$ taking \nthe place of the $\\g{\\mu}$'s of that previous calculation. The function of the $\\g5$ is to ensure that the \npartial derivatives become restricted. If the gauge group had been non-abelian, then it would also \nensure that we obtain commutators, not anticommutators, of all the boson fields.\n\n\n\\section{Non-abelian Yang-Mills theory with mixed gauge fields}\n\\noindent\n\tConsider a lagrangian that transforms under a non-abelian local Lie group that has \n$N$ generators. The fermion or matter sector of the non-abelian lagrangian has the form $\\overline{\\psi} i \\hbox{$\\not \\! \\! D$} \\psi$, \nwhere $D_\\mu$ is a covariant derivative chosen to maintain gauge invariance. This term is invariant \nunder the transformation $\\psi \\to U \\psi$, where $U=U(x)$ is an element of the fundamental \nrepresentation of the group. The covariant derivative is $D_\\mu = \\deriv{\\mu} + A_\\mu$, where $A_\\mu = ig A_\\mu^{\\ a} (x) T^a$ is \nan element of the Lie algebra and $g$ is a coupling constant. We are assuming here that the set of \nmatrices $\\{ T^a \\}$ is a representation of the group's generators. Gauge invariance of the matter term \nis assured if\n\\ecuacion{\nA_\\mu \\to U A_\\mu U^{-1} - (\\deriv{\\mu} U) U^{-1}\n},\nor, what is the same,\n\\ecuacion{\n\\hbox{$\\not \\! \\! A$} \\to U \\hbox{$\\not \\! \\! A$} U^{-1} - (\\hbox{$\\not \\! \\partial$} U) U^{-1}\n}.\n\n\tWe have already seen how scalar fields can function as gauge fields. Our aim in \nthis section is to construct a non-abelian theory that uses both scalar and vector gauge fields. We \nproceed as follows. For every generator in the Lie group we choose one gauge field, it does not \nmatter whether vector or scalar. As an example, suppose there are $N$ generators in the Lie group; \nwe choose the first $N_V$ to be associated with an equal number of vector gauge fields and the last \n$N_S$ to be associated with an equal number of scalar gauge fields. Naturally $N_V + N_S = N$. Now we \nconstruct a covariant derivative $D$ by taking each one of the generators and multiplying it by one \nof its associated gauge fields and summing them together. The result is\n\\ecuacion{\nD \\equiv \\hbox{$\\not \\! \\partial$} + \\hbox{$\\not \\! \\! A$} + \\Phi\n\\label{general}\n},\n\\ecuarreglo{\n\\hbox{$\\not \\! \\! A$} & \\equiv & \\g{\\mu} A_\\mu \\equiv ig \\g{\\mu} A_\\mu^{\\ a} T^a \\coma, \\qquad \\qquad a = 1, \\ldots, N_V \\ptc,\n\\Phi    & \\equiv & \\g5 \\varphi \\equiv -g \\g5 \\varphi^b T^b \\coma, \\qquad \\qquad b = N_V+1, \\ldots, N  \\nonumber\n}.\n\nNotice the difference between $A_\\mu$ and $A_\\mu^{\\ a}$, and between $\\varphi$ and $\\varphi^b$. We take the gauge \ntransformation for these fields to be\n\\ecuacion{\n\\hbox{$\\not \\! \\! A$} + \\Phi \\to U (\\hbox{$\\not \\! \\! A$} + \\Phi) U^{-1} - (\\hbox{$\\not \\! \\partial$} U) U^{-1}\n},\nfrom which one can conclude that\n\\ecuacion{\nD \\to U D U^{-1}\n}.\n\nThe following lagrangian is constructed based on the requirements that it contains only matter \nfields and covariant derivatives, and that it possesses both Lorentz and gauge invariance:\n\\ecuacion{\n{\\cal L}_{NA} = \\overline{\\psi} i D \\psi + \\frac{1}{2g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\frac{1}{8} \\mathop {\\rm Tr}\\nolimits^2 D^2 - \\frac{1}{2} \\mathop {\\rm Tr}\\nolimits D^4 \\right)\n\\label{lagNA}\n},\nwhere the trace with the tilde is over the Lie group matrices and the one without it is over \nmatrices of the spinorial representation of the Lorentz group. The additional factor of $1\/2$ that the \ntraces of \\refeq{lagNA} have with respect to \\refeq{Teo} comes from normalization \\refeq{norma}, that is the usual \none in the non-abelian case.\n\n\tAlthough lagrangian ${\\cal L}_{NA}$ was constructed based only on the requirements just \nmentioned, it is an interesting fact that the expansion of the covariant derivative into its \ncomponent fields results in expressions that are traditional in Yang-Mills theories. The reader who \nwishes to make the expansion herself can substitute \\refeq{general} in \\refeq{lagNA}, keeping in mind the derivatives \nare unrestrained, and aim first for the intermediate result\n\\ecuarreglo{\n\\frac{1}{16} \\mathop {\\rm Tr}\\nolimits^2 D^2 - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits D^4 &=& \\left( (\\partial \\cdot A)+A^2 \\right)^2 - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits \\left( (\\hbox{$\\not \\! \\partial$} \\hbox{$\\not \\! \\! A$})+\\hbox{$\\not \\! \\! A$} \\hbox{$\\not \\! \\! A$} \\right)^2 \\nonumber \\\\\n   & & - \\frac{1}{4} \\mathop {\\rm Tr}\\nolimits \\left( (\\hbox{$\\not \\! \\partial$} \\Phi)+\\{ \\hbox{$\\not \\! \\! A$} \\Phi \\} \\right)^2\n\\label{expandido}\n},\nwhere the curly brackets denote an anticommutator. (We recommend to use here the same trick \nexplained in section 2.) Notice in this expression that the differentiation operators are restrained, \nand that the two different types of gauge fields appear in an anticommutator. One of the effects of \nthe $\\g5$ in \\refeq{general} is to turn this anticommutator into a commutator through the use of the properties \nof Clifford algebras. Substituting \\refeq{general} in \\refeq{expandido} and in the matter term of \\refeq{lagNA} we obtain the \nnon-abelian lagrangian in expanded form:\n\\ecuarreglo{\n{\\cal L}_{NA} &=& \\overline{\\psi} i(\\hbox{$\\not \\! \\partial$} + \\hbox{$\\not \\! \\! A$}) \\psi - g \\overline{\\psi} i \\g5 \\varphi^b T^b \\psi + \\frac{1}{2 g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\deriv{\\mu} A_\\nu - \\deriv{\\nu} A_\\mu + [A_\\mu, A_\\nu] \\right)^2 \\nonumber \\\\\n   & & + \\frac{1}{g^2} \\widetilde{\\mathop {\\rm Tr}\\nolimits} \\left( \\deriv{\\mu} \\varphi + [A_{\\mu}, \\varphi] \\right)^2\n\\label{lagexpandido}\n}.\n\nThe reader will recognize familiar structures: the first term on the right looks like the usual matter \nterm of a gauge theory, the second like a Yukawa term, the third like the kinetic energy of vector \nbosons in a Yang-Mills theory and the fourth like the gauge-invariant kinetic energy of scalar \nbosons in the non-abelian adjoint representation. It is also interesting to observe that, if in the last \nterm we set the vector bosons equal to zero, then the term simply becomes $\\sum_{b,\\mu} \\frac{1}{2} \\deriv{\\mu}\\varphi^b \\partial^\\mu \\varphi^b$, the \nkinetic energy of the scalar bosons. We have constructed a generic non-abelian gauge theory with \ngauge fields that can be either scalar or vector.\n\n\n\\section{The GWS model using \\bfsu3\\ and \\bfuu3}\n\\noindent\n\tThe obvious choice for a small group to simplify and unify the GWS model using a \ngeneralized gauge theory is \\su3, because it has the correct hypercharge numbers for the leptons \nof a chiral multiplet. Furthermore, it has 8 generators, while the GWS model has precisely 8 \nbosons: 4 vector ones and the 4 Higgs real scalar fields. Unfortunately, this choice does not work \nas we shall soon see. Let $A_\\mu^{\\ a}$, $a=1,2,3$, and $B_\\mu$ be four vector fields to which we associate the \nfour Gell-Mann generators $\\ld{a}$, $a=1,2,3$, and $\\ld{8}$, respectively. Let $\\varphi^b$, $b=4,5,6,7$, be four real \nscalar fields, to which we associate the remaining Gell-Mann generators $\\ld{b}$, $b=4,5,6,7$. The \ncovariant derivative can be found following the prescription given in \\refeq{general}. Using the usual \nrepresentation of the Gell-Mann matrices and $g$ as coupling constant it can be explicitly written \nas follows:\n\\ecuacion{\nD = \\hbox{$\\not \\! \\partial$} + \\frac{g}{2} \\pmatrix{i \\hbox{$\\bf \\not \\! \\! A$} \\cdot \\hbox{{\\boldmath $\\sigma$}} + i \\hbox{$\\not \\! \\! B$} {\\bf 1} \/ \\sqrt{3} & -\\sqrt{2} \\g5 \\widehat{\\varphi} \\cr\n-\\sqrt{2} {\\widehat{\\varphi}}^{\\dag} \\g5 & -2i \\hbox{$\\not \\! \\! B$} \/ \\sqrt{3} }\n\\label{DSU3}\n},\nwhere the $\\sigma^a$, $a=1,2,3$, are the Pauli matrices, ${\\bf 1}$ is a $2 \\times 2$ matrix, and\n\\ecuacion{\n\\widehat{\\varphi} = \\frac{1}{\\sqrt{2}} \\pmatrix{ \\varphi^4 - i \\varphi^5 \\cr \\varphi^6 - i \\varphi^7 }\n}.\n\nThe lagrangian of the model is then precisely ${\\cal L}_{NA}$, as expressed by \\refeq{lagNA}, with the covariant \nderivative given by \\refeq{DSU3} and the fermion triplet by $\\psi_L$, the chiral triplet of equation \\refeq{chiral}. To \nobtain the expanded form of the boson kinetic energy sector through straightforward calculation \nis very messy, but through the use of formula \\refeq{lagexpandido} it is possible to arrive at the following \nexpression without much trouble:\n\\ecuarreglo{\n{\\cal L}_{NA} &=&\\overline{\\theta_L} (i\\hbox{$\\not \\! \\partial$} - \\frac{1}{2} g \\hbox{$\\not \\! \\! A$}^a \\sigma^a - \\frac{1}{2} g' \\hbox{$\\not \\! \\! B$}) \\theta_L + \\overline{e_R} (i \\hbox{$\\not \\! \\partial$} - g' \\hbox{$\\not \\! \\! B$}) e_R \\nonumber \\\\\n          & &+i \\frac{g}{\\sqrt{2}} \\overline{e_L^c} \\widehat{\\varphi}^{\\dag} \\theta_L +i \\frac{g}{\\sqrt{2}} \\overline{\\theta_L} \\widehat{\\varphi} e_L^c \\nonumber \\\\\n          & &-\\frac{1}{4} {\\bf A_{\\mu \\nu} \\cdot A^{\\mu \\nu}} -\\frac{1}{4} B_{\\mu \\nu} B^{\\mu \\nu} \\nonumber \\\\\n          & &+\\left| (\\deriv{\\mu} +\\frac{1}{2} ig A_\\mu^{\\ a} \\sigma^a + \\frac{3}{2} ig' B_\\mu) \\widehat{\\varphi} \\right|^2\n\\label{lagSU3}\n},\nwhere $\\theta_L = (\\nu_L, e_L)^T$ and we have introduced the symbol $g'\\equiv g\/\\sqrt{3}$. This is \\letra{almost} the lagrangian \nof the GWS model for a Weinberg angle $\\theta_W = 30^\\circ$ (a value very close to the experimental one), \nexcept that the hypercharge of the Higgs comes out as $3$ and not $-1$ as it should be. This detail \ndooms the model. Notice too the Yukawa terms are actually null, as one can see from chirality \nconsiderations.\n\n\tLet us look at the problem of the hypercharge in more detail. Let\n\\ecuacion{\n\\ld{Y} = \\frac{1}{\\sqrt{3}} \\mathop {\\rm diag} (x,x,y)\n}{}\nbe some generator associated with the vector field $B_\\mu$, the one that eventually becomes the \nisospin singlet in the GWS model; then the hypercharge of the Higgs is $Y=x-y$. In the example \nabove we were using $\\ld{8}$, shown in \\refeq{gene8}, so that $Y=1-(-2)=3$. If instead of \\su3\\ we had been \nusing $SU(2\/1)$\\ then the choice for $\\ld{Y}$ would have been $\\ld{0}$, shown in \\refeq{gene0}, and $Y=1-(+2)=-1$, the \ncorrect value. So it seems we have reached an \\letra{impasse}: the correct hypercharge is given precisely \nby the group we do not want to use. The way out of it is to realize that a very similar group, \n\\uu3, has a representation where $\\ld{0}$ appears. In other words, it is not necessary to go to graded \ngroups to obtain the correct hypercharge. We will see now how this comes about.\n\n\tA representation of \\uu3\\ is, for instance, the 8 Gell-Mann generators $\\ld{a}$, $a=1,\\ldots ,8$ plus \nanother (not traceless) $3 \\times 3$ matrix which we take to be the normalized unit matrix:\n\\ecuacion{\n\\ld{9} = \\sqrt{\\frac{2}{3}} \\mathop {\\rm diag} (1,1,1)\n}.\n\nWe now define a new matrix\n\\ecuacion{\n\\ld{10} = \\sqrt{\\frac{2}{3}} \\mathop {\\rm diag} (1,1,-1)\n},\nand make the observation that it and $\\ld{0}$ can be expressed as linear combinations of $\\ld{8}$ and $\\ld{9}$:\n\\ecuarreglo{\n\\ld{10} &=& \\frac{2 \\sqrt{2}}{3} \\ld{8} + \\frac{1}{3} \\ld{9} \\ptc,\n\\ld{0} &=& -\\frac{1}{3} \\ld{8} + \\frac{2 \\sqrt{2}}{3} \\ld{9}\n}.\nNotice that $\\ld{10}$ and $\\ld{0}$ are orthonormal under \\refeq{norma}. The set of 9 matrices $\\ld{10}$, $\\ld{0}$ and $\\ld{a}$, \n$a=1,\\ldots ,7$ forms a representation of \\uu3, since each matrix is a linear combination of the \ngenerators of another representation. So we have found $\\ld{0}$ in \\uu3, and associating it with $B_\\mu$ we \nmake sure that the hypercharge for the Higgs comes out correctly.\n\n\tThis new group has 9 generators, and, therefore, 9 bosons. This is cause for concern, since \nthe extra boson could upset the precise clockwork phenomenology of the GWS model. We have \nin principle the option of taking it to be either scalar or vector, but the second option would \npresent us with an unphysical extra vector boson. On the other hand, if we take it to be a scalar \nboson, it is an interesting fact that the boson completely decouples from the rest of the particles \nand we obtain again lagrangian \\refeq{lagSU3}, but with the correct hypercharge for the Higgs bosons. Now \n$\\psi$ is given by \\refeq{nonchiral} and not by \\refeq{chiral}, as in the \\su3\\ case.\n\n\tTo understand why the new scalar boson, that we shall call $\\Upsilon$, decouples, go back to \\refeq{lagexpandido}, \nthe non-abelian lagrangian, and notice that the scalar bosons only appear in the second and fourth \nterms of the right of the equation. Since the scalar boson is associated with $\\ld{10}$, which is a \ndiagonal generator, in the second term the spinorial degrees of freedom are all multiplying their \nrespective conjugates, that have the opposite chirality, and therefore all products are null. In the \nfourth term within the parenthesis there appears, for the case of the new scalar, the expression $[A_\\mu, \\ld{10} \\Upsilon]$,\nthat is, the commutator of a block diagonal matrix and $\\ld{10}$, and, therefore, zero. The \nonly term that remains in the lagrangian that contains $\\Upsilon$ is its kinetic energy. It would seem this \nfield is massless, as it does not couple with the Higgs.\n\n\tWe again expand ${\\cal L}_{NA}$, this time using the new set of generators and the nonchiral \nfermion triplet $\\psi$ of equation \\refeq{nonchiral}, with the result:\n\\ecuarreglo{\n{\\cal L}_{NA} &=&\\overline{\\theta_L} (i\\hbox{$\\not \\! \\partial$} - \\frac{1}{2} g \\hbox{$\\not \\! \\! A$}^a \\sigma^a - \\frac{1}{2} g' \\hbox{$\\not \\! \\! B$}) \\theta_L + \\overline{e_R} (i\\hbox{$\\not \\! \\partial$} - g' \\hbox{$\\not \\! \\! B$}) e_R \\cr\n          & &+i \\frac{g}{\\sqrt{2}} \\overline{e_R} \\widehat{\\varphi}^{\\dag} \\theta_L -i \\frac{g}{\\sqrt{2}} \\overline{\\theta_L} \\widehat{\\varphi} e_R \\cr\n          & &-\\frac{1}{4} {\\bf A_{\\mu \\nu} \\cdot A^{\\mu \\nu}} -\\frac{1}{4} B_{\\mu \\nu} B^{\\mu \\nu} \\cr\n          & &+\\left| (\\deriv{\\mu} +\\frac{1}{2} ig A_\\mu^{\\ a} \\sigma^a - \\frac{1}{2} ig' B_\\mu) \\widehat{\\varphi} \\right|^2 + \\frac{1}{2} (\\deriv{\\mu} \\Upsilon)(\\partial^\\mu \\Upsilon)\n\\label{lagU3}\n},\nwhich is the lagrangian of the GWS model. Notice the two Yukawa terms do not vanish in this \nlagrangian, as they did in \\refeq{lagSU3}. The difference of sign between them does not matter, as it simply \ndepends on what phase of the $e_R$\\ we decide to use.\n \n\tIn Yang-Mills theories when the covariant derivative acts on a field $X$, its gauge fields \nacquire certain coefficients called the ``charges'' of each gauge field with respect to $X$. In this \ngeneralized gauge theory the same thing has to be done. Thus, when $D$ acts on the leptonic \ntriplet, its gauge fields are going to be multiplied by constants we shall call ``leptonic charges'' $Q_V$ \nand $Q_S$, with the result $D \\psi = (\\hbox{$\\not \\! \\partial$} + Q_V \\hbox{$\\not \\! \\! A$} + Q_S \\Phi) \\psi$. From our knowledge of the standard model we \nconclude that $Q_V=1$, and that there are three $Q_S$, one for each generation, and have rather small \nvalues. Lagrangian \\refeq{lagU3} is actually for one generation only, and it should include its respective $Q_S$ \nas a coefficient in the two Yukawa terms. We have no \\letra{a priori} knowledge of the values of the \nleptonic charges.\n\n\n\\section{Summary and remarks}\n\\noindent\n\tWe found a particular way of expressing the kinetic energy of vector bosons (what in \nprincipal vector bundles is called the curvature) so that those fields appear contracted only with \nDirac matrices. From here we were able to generalize the concept of covariant derivative, so that \nit included both scalar and vector bosons. Using this generalized derivative one can write the \nGWS model using only two terms (a curvature and a matter term), and unify the two groups (and \ntheir coupling constants) into one. This derivative has to transform as a slashed 4-vector.\n\n\tThe unification group is not \\su3, that gives the wrong value for the hypercharge of the \nHiggs field, but \\uu3\\ instead, that predicts correctly all the quantum numbers of the GWS model. \nThis is possible because there is a representation of this group that contains the same generator of \n$SU(2\/1)$\\ that gives the right hypercharge to the Higgs bosons. Since we did not use a graded \ngroup, the Higgs fields obey, correctly, Bose-Einstein statistics, and the kinetic energy terms of \n\\letra{all} vector bosons have the right sign.\tAn extra scalar boson has to included (since \\uu3\\ has one \nmore generator than \\su3) but it automatically decouples from the rest of the model becoming \nan unobservable particle. It is probably massless, since it does not couple with the Higgs field, \neither.\n\nIn the GWS model symmetry breaking is achieved spontaneously through the\nintroduction of a potential of the form $V(\\varphi)$. In the model presented here \nthis potential also has to be introduced explicitly, as it does not appear in its\nsole\ntwo terms, kinetic energy of bosons and of fermions. In this it differs from the\nnoncommutative geometry results for the standard model, that implicitly include\nthis potential. The terms for the Higgs potential appear in our model, but they \ncancel since we introduce counterterms for the purpose of avoiding the presence of\ndifferentiation operators acting indefinitely to the right. (See equation\n\\refeq{Teo}.) These counterterms also avoid other unwanted self-interacting terms.\n\nThe Yukawa coupling constants appear in a natural way as generalized gauge charges\nof the lepton triplet $\\psi$, but the model sheds no light about what those values\nmay be, or why there is a different value for each generation.\n\n\\nonumsection{References}\n\\noindent\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Keywords:} Social Robotics, Autism, Music Therapy, Turn-Taking, Motor Control, Emotion Classification}\n\t\\end{abstract}\n\t\n\t\\section{Introduction}\n\t\\ Throughout history, music has been used medicinally due to its notable impact on the mental and physical health of its listeners. This practice became so popular over time that it ultimately transitioned into its own type of therapy called music therapy. The interactive nature of music therapy has also made it especially well-suited for children. Children's music therapy is performed either in a one-on-one session or a group session and by listening, singing, playing instruments, and moving, patients can acquire new knowledge and skills in a meaningful way. It has previously been shown to help children with communication, attention, and motivation problems as well as behavioral issues \\cite{cibrian2020supporting,gifford2011using}.\n\tWhile music therapy is extremely versatile and has a broad range of applications, previous research studies have found  that using music as an assistive method is especially useful for children with autism \\cite{cibrian2020supporting,mossler2019therapeutic, dellatan2003use,brownell2002musically,starr1998understanding}. This interest in music therapy as a treatment for children with autism has even dated as far back as the early $20^{th}$ century. In fact, in the 1940s, music therapy was used at psychiatric hospitals, institutions, and even schools for children with autism. Due to the fact both the diagnostic criteria for autism and music therapy as a profession were only just emerging at this time, no official documentation or research about this early work exists. In the next decade, the apparent unusual musical abilities of children with autism intrigued many music therapists \\cite{warwick1991music}. By the end of the 1960s, music therapists started delineating goals and objectives for autism therapy \\cite{reschke2011history}. Since the beginning of the 1970s and onward,  theoretically grounded music therapists have been working toward a more clearly defined approach to improve the lives of individuals with autism spectrum disorders (ASD). However, for decades, music therapists have not used a consistent assessment method when working with individuals with ASD. A lack of a quality universal assessment tool has caused therapy to drift towards non-goal driven treatment and has reduced the capacity to study the efficacy of the treatment in a scientific setting \\cite{thaut2000scientific}.\n\t\n\tHumanoid socially assistive robots are ideal to address some of these concerns \\cite{scassellati2012robots, diehl2012clinical}. Precise programming can be implemented to ensure the robots deliver therapy in a consistent assessment method each session. Social robots are also unique in their prior success in working with children with ASD \\cite{boucenna2014interactive, pennisi2016autism}. Previous studies have found children with autism have less interest in communicating with humans due to how easily these interactions can become overwhelming \\cite{marinoiu20183d,di2018deep,richardson2018robot,feng2013can} and instead are more willing to interact with humanoid social robots in daily life due to their relatively still faces and less intimidating characteristics. Some research has found that children with ASD speak more while interacting with a non-humanoid robot compared to regular human-human interactions \\cite{kim2013social}. A ball-like robot called Sphero has been used to examine different play patterns and emotional responses of children with ASD \\cite{boccanfuso2016emotional, boucenna2014learning}. Joint attention with body movement has also been tested and evaluated using a robot called NAO from \\cite{anzalone2014children}. Music interaction has been recently introduced in Human-Robot-Interaction (HRI) as well \\cite{taheri2019teaching}. The NAO robot has been widely used in this field such as music and dance imitation \\cite{beer2016robot} and body gesture imitation \\cite{guedjou2017influence,zheng2015robot,boucenna2016robots}.\n\tTo expand upon the aforementioned implications of socially assistive robotics and music, \\textbf{the scientific question of this research is whether music-therapy delivered by a socially assistive robot is engaging and effective enough to serve as a viable treatment option for children with ASD}.\n\tThe main contribution of this paper is as follows. First, we propose a fully autonomous assistive robot-based music intervention platform for children with autism. Second, we designed HRI sessions and conducted studies based on the current platform, where motor control and turn-taking skills were practiced by a group of children with ASD. Third, we utilized and integrated machine learning techniques into our music-therapy HRI sessions to recognize and classify event-based emotion expressed by the study participants.\n\t\n\tThe remainder of this paper is organized as follows. Section 2 presents related works concerning human-robot interaction in multiple intervention methods. Section 3 elaborates on the experiment design process including the details of the hardware and HRI sessions. We present our music therapy platform in Section 4 and the experimental results in Section 5. Finally, Section 6 concludes the paper with remarks for future work.\n\t\n\t\\section{Related Works}\n\t\n\t\\ Music is an effective method to involve children with autism in rhythmic and non-verbal communication and has often been used in therapeutic sessions with children who suffer from mental and behavioral disabilities \\cite{lagasse2019assessing,boso2007effect,roper2003melodic}. Nowadays, at least 12\\% of all treatments for individuals with autism consist of music-based therapies \\cite{bhat2013review}. Specifically, playing music to children with ASD in therapy sessions has shown a positive impact on improving social communication skills \\cite{lagasse2019assessing,lim2011effects}. Many studies have utilized both recorded and live music in interventional sessions for single and multiple participants \\cite{dvir2020body,bhat2013review, corbett2008brief}. Different social skills have been targeted and reported (i.e., eye-gaze attention, joint attention and turn-taking activities) in music-based therapy sessions \\cite{stephens2008spontaneous, kim2008effects}. Improving gross and fine motor skills for children with ASD through music interventions is  noticeably absent in this field of studies \\cite{bhat2013review} and thus is one of the core features in the proposed study.\n\tSocially assistive robots are becoming increasingly popular as interventions for youth with autism. Previous studies have focused on eye contact and joint attention \\cite{mihalache2020perceiving,  mavadati2014comparing,feng2013can}, showing that the pattern of gaze perception in the ASD group is similar to Typically Developing (TD) children as well as the fact eye contact skills can be significantly improved after interventional sessions. These findings also provide strong evidence that ASD children are more inclined to engage with humanoid robots in various types of social activities, especially if the robots are socially intelligent \\cite{anzalone2015evaluating}. Other researchers have begun to use robots to conduct music-based therapy sessions. In such studies, children with autism are asked to imitate music based on the \\textit{Wizard of Oz} and Applied Behavior Analysis (ABA) models using humanoid robots in interventional sessions to practice eye-gaze and joint attention skills \\cite{askari2018pilot,taheri2015impact, taheri2016social}.\n\t\n\tHowever, past research has often had certain disadvantages, such as a lack of an automated system in human-robot interaction. This research attempts to address these shortcomings with our proposed platform. In addition, music can be used as a unique window into the world of autism as a growing body of evidence suggests that individuals with ASD are able to understand simple and complex emotions in childhood using music-based therapy sessions \\cite{molnar2012music}. Despite the obvious advantages of using music-therapy to study emotion comprehension in children with ASD, limited research has been found, especially studies utilizing physiological signals for emotion recognition in ASD and TD children \\cite{feng2018wavelet}. The current study attempted to address this gap by using electrodermal activity (EDA) as a measure of emotional arousal to better understand and explore the relationship between activities and emotion changes in children with ASD. To this end, the current research presents an automated music-based social robot platform with an activity-based emotion recognition system. The purpose of this platform is to provide a possible solution for assisting children with autism and help improve their motor and turn-taking skills. Furthermore, by using bio-signals with Complex-Morlet (C-Morlet) wavelet feature extraction \\cite{feng2018wavelet}, emotion classification and emotion fluctuation can be analyzed based on different activities. TD children are included as a control group to compare the intervention results with the ASD group.\n\t\n\t\\section{Experiment Design}\n\t\\subsection{Participants Selection}\n\tNine high functioning ASD participants (average age: 11.73, std: 3.11) and seven TD participants (average age: 10.22, std: 2.06) were recruited for this study. Only one girl was included in the ASD group, which according to \\cite{loomes2017male}, is under the estimated ratio of 3:1 for male and females with ASD. ASD participants who had participated in previous unrelated research studies in the past were invited to return for this study. Participants were selected from this pool and the community with help from the University of Denver Psychology department. Six out of nine ASD participants had previous human-robot-interaction with another social robot named ZENO \\cite{mihalache2020perceiving}, and three out of the nine kids from the ASD group had previous music experience on instruments other than the Xylophone (saxophone and violin). All children in the ASD group had a previous diagnosis of ASD in accordance with diagnostic criteria outlined in \\cite{DSMIV2000}, including an ADOS report on record. Additionally, their parents completed the Social Responsiveness Scale (SRS) \\cite{constantino2012social} for their child. The SRS provides a quantitative measure of traits associated with autism among children and adolescents between four and eighteen years-of-age. All ARS T-scores in our sample were above 65. Unlike the ASD group, TD kids had no experience with NAO or other robots prior to this study. Most of them, on the other hand, did participate in music lessons previously. Experimental protocols were approved by the University of Denver Institutional Review Board.\n\t\n\t\\subsection{NAO: A Humanoid Robot}\n\tNAO, a humanoid robot merchandised by SoftBank Group Corporation, was selected for the current research. NAO is 58 cm (23 inches) tall and has 25 degrees of freedom, allowing him to perform most human body movements. According to the official Aldebaran manufacturer documentation, NAO's microphones have a sensitivity of 20mV\/Pa +\/-3dB at 1kHz and an input frequency range of 150Hz - 12kHz. Data is recorded as a 16 bit, 48000Hz, 4 channel wave file which meets the requirements for designing the online feedback audio score system. NAO's computer vision module includes face and shape recognition units. By using the vision feature of the robot, NAO can see an instrument with its lower camera and implement an eye-hand self-initiative system that allows the robot to micro-adjust the coordination of its arm joints in real-time, which is very useful to correct cases of inproper positioning prior to engaging with the Xylophone.\n\t\n\tNOA's arms have a length of approximately 31 cm. Position feedback sensors are equipped in each of the robot's joints to obtain real-time localization information. Each robot arm has five degrees of freedom and is equipped with sensors to measure the position of joint movement. To determine the position of the Xylophone and the mallets' heads, the robot analyzed images from the lower monocular camera located in its head, which has a diagonal field of view of 73 degrees. By using these dimensions, properly sized instruments can be selected and more accessories can be built.\n\t\n\t\\subsection{Hardware Accessories}\n\tIn order to have a well-functioning toy-size humanoid robot play music for children with autism, some necessary accessories needed to be made before the robot was capable of completing this task. All accessories will be discussed in the following paragraphs.\n\t\n\tIn this system, due to the length of NAO's outstretched arms, a Sonor Toy Sound SM Soprano Glockenspiel with 11 sound bars 2cm in width was selected and utilized in this research. The instrument is $31cm \\times 9.5cm \\times 4cm$, including the resonating body.\n\tThe smallest sound bar is playable in an area of $2.8cm \\times 2cm$, the largest in an area of $4.8cm \\times 2cm$. The instrument is diatonically tuned in C major\/A minor. The 11 bars of the Xylophone represent 11 different notes, or frequencies, that cover one and a half octave scales, from C6 to F7.\n\tThe Xylophone, also known as the marimba or the glockenspiel, is categorized as a percussion instrument that consists of a set of metal\/wooden bars that are struck with mallets to produce delicate musical tones. Much like the keyboard or drums, to play the Xylophone properly, a unique and specific technique needs to be applied. A precise striking movement is required to produce a beautiful note, an action perfect for practicing motor control, and the melody played by the user can support learning emotions through music.\n\t\n\tFor the mallets, we used the pair that came with the Xylophone, but added a modified 3D-printed gripper that allowed the robot hands to hold them properly (see Figure \\ref{griper}). The mallets are approximately 21 cm in length and include a head with a 0.8 cm radius. Compared to other designs, the mallet gripper we added encourages a natural holding position and allows the robot to properly model how participants should hold the mallet stick.\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/grip.eps, scale = 0.4}\\label{griper} \\\\\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Mallet Griper} \\label{griper}\n\t\t\\end{center}\n\t\\end{figure}\n\tUsing carefully measured dimensions, a wooden base was designed and laser cut to hold the Xylophone at the proper height for the robot to play in a crouching position. In this position, the robot could easily be fixed in a location and have the same height as the participants, making it more natural for the robot to teach activities (see Figure \\ref{stand}).\\\\\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/front_view.eps, scale = 0.4} \\label{front}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Instrument Stand Front View.} \\label{stand}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsection{Q-Sensor}\n\tOne Q-sensor \\cite{kappas2013validation} was used in this study. Participants were required to wear this device during each session.  EDA signal\n\t(frequency rate 32Hz) was collected from the Q-sensor attached to the wrists (wrist side varied and was determined by the participants). Due to the fact participants often required breaks during sessions, 2 to 3 seperated EDA files were recorded. These files were annotated by comparing the time stamps with the videos manually.\n\t\n\t\\subsection{Experiment Room Setup}\n\tAll experiment sessions were held in an $11ft \\times 9.5ft \\times 10ft$ room with six HD surveillance cameras installed in the corners, on the sidewalls, and on the ceiling of the room (see Figure \\ref{room}).The observation room was located behind a one-way mirror and participants were positioned with their backs toward this portion of the room to avoid distractions. During experiment sessions, an external, hand-held audio recorder was set in front of participants to collect high-quality audio to use for future research.\n\t\\begin{figure*}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\epsfig{figure=.\/fig\/room_pana.eps, scale = .25}\\label{room} \n\t\t\t\\end{tabular}\n\t\t\t\\caption{Experiment room.} \\label{room}\n\t\t\\end{center}\n\t\\end{figure*}\n\t Utilizing a miniature microphone attatched to the ceiling camera, real-time video and audio were broadcasted to the observation room during sessions, allowing researchers to observe and record throughout the sessions. Parents or caregivers in the observation room could also watch and call off sessions in the case of an emergency.\n\t \n\t\\subsection{Experiment Sessions}\n\tFor each participant in the ASD group, six total sessions were delivered, including one baseline session, four interventional sessions, and one exit session. Only the baseline and exit sessions were required for the TD group. Baseline and exit sessions contained two activities: 1) music practice and 2) music gameplay. Intervention sessions contained three parts: S1) warm-up; S2) single activity practice (with a color hint); and S3) music gameplay. Every session lasted for a total of 30-60 minutes depending on the difficulty of each session and the performance of individuals. Typically, each participant had baseline and exit session lengths that were comparable. For intervention sessions, the duration gradually increased in accordance with the increasing difficulty of subsequent sessions. Additionally, a user-customized song was used in each interventional session to have participants involved in multiple repetitive activities. The single activity practice was based on music practice from the baseline\/exit session. In each interventional session, the single activity practice only had one type of music practice. For instance, single-note play was delivered in the first intervention session. For the next intervention session, the single activity practice increased in difficulty to multiple notes. The level of difficulty for music play was gradually increased across sessions and all music activities were designed to elicit an emotional reaction.\n\t\n\tTo aid in desired music-based social interaction, NAO delivered verbal and visual cues indicating when the participant should take action. A verbal cue consisted of the robot prompting the participant with \"Now, you shall play right after my eye flashes.\" Participants could also reference NAO's eyes changing color as a cue to start playing. Aftter the cues, participants were then given five to 10 seconds to replicate NAO's strikes on the Xylophone. The start of each session was considered a warmup and participants were allowed to imitate NAO freely without accuracy feedback. The purpose of having a warm-up section was to have participants focus on practicing motor control skills while also refreshing their memory on previous activities. Following the warmup, structured activities would begin with the goal of the participant ultimately completing a full song. Different from the warmup section, notes played during this section in the correct sequence were considered to be a good-count strike and notes incorrectly played were tracked for feedback and additional training purposes. To learn a song, NAO gradually introduced elements of the song, starting with a single note and color hint. Subsequently, further notes were introduced, then the participants were asked to play half the song, and then finally play the full song after they perfected all previous tasks. Once the music practice was completed, a freshly designed music game containing three novel entertaining game modes was presented to the participants. Participants could then communicate with the robot regarding which mode to play with. Figure \\ref{interact} shows the complete interaction process in our HRI study.\n\t\n\t\\begin{figure*}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\t\\epsfig{figure=.\/fig\/interact.eps, scale = .45}\\label{r} \n\t\t\t\\end{center}\n\t\t\t\\caption{Experiment session illustration.} \\label{interact}\n\t\t\\end{center}\n\t\\end{figure*}\n\t\n\tThe laboratory where these experiments took place is 300-square feet and has environmental controls that allowed the temperature and humidity of the testing area to be kept consistent. For experimental purposes in the laboratory, the ambient temperature and humidity were kept constant to guarantee the proper functionality of the Q-sensor as well as the comfort of subjects in light clothing. To obtain appropriate results and ensure the best possible function of the device, the sensor was cleaned before each usage. Annotators were designated to determine the temporal relation between the video frames and the recorded EDA sequences of every participant. To do this, annotators went through video files of every session, frame by frame, and designated the initiation and conclusion of an emotion. Corresponding sequences of EDA signals were then identified and utilized to create a dataset for every perceived emotion. \n\t\n\tFigure \\ref{eda_anno} shows the above-described procedure diagrammatically. Due the fact that it is hard to conclude these emotions with specific facial expressions, event numbers will be used in the following analysis section representing emotion comparison.  The rest of the emotion labels respresent the dominant feeling of each music activity according to the observations of both annotators and the research assistant who ran the experiment sessions. \n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/fig5.eps}\\label{eda_anno}\n\t\t\t\\end{center}\n\t\t\t\\caption{The distribution of the targeted emotions across all subjects and events. \n\t\t\t} \\label{eda_anno}\n\t\t\\end{center}\n\t\\end{figure}\n\n\t\\;\\noindent\\textbf{Mode 1):} The robot randomly picks a song from its song bank and plays it for participants. After each song, participants were asked to identify a feeling they ascribed to the music to find out whether music emotion can be recognized. \n\t\n\t\\textbf{Mode 2):} A sequence of melodies were randomly generated by the robot with a consonance (happy or comfortable feeling) or dissonance (sad or uncomfortable feeling) style. Participants were asked to express their emotions verbally and a playback was required afterwards.\n\t\n\t\\textbf{Mode 3):} Participants have five seconds of free play then challenge the robot to imitate what was just played. After the robot was done playing, the participant rated the performance, providing a reversed therapy-like experience for all human volunteers. \n\t\n\tThere was no limit on how many trials or modes each individual could play for each session, but each mode had to be played at least once in a single session. The only difference between the baseline and exit sessions was the song that was used in them. In the baseline session, \"Twinkle, Twinkle, Little Star\" was used as an entry-level song for every participant. For the exit session, participants were allowed to select a song of their choosing so that they would be more motivated to learn the music. In turn, this made the exit session more difficult than the baseline session. By using the Module-Based Acoustic Music Interactive System, inputting multiple songs became possible and less time-consuming. More than 10 songs were collected in the song bank including \"Can Can\" by Offenbach, \"Shake It Off\" by Taylor Swift, the \"SpongeBob SquarePants\" theme from the animated show SpongeBob SquarePants, and \"You Are My Sunshine\" by Johnny Cash. Music styles covered kid's songs, classic, pop, ACG (Anime, Comic and Games), and folk, highlighting the versatile nature of this platform. In addition to increasing the flexibility of the platform, having the ability to utilize varying music styles allows the robot to accommodate a diverse range of personal preferences, further motivating users to learn and improve their performance.\n\t\n\t\\section{Module-Based Acoustic Music Interactive System Design}\n\tIn this section, a novel, module-based robot-music therapy system will be presented. For this system to be successful, several tasks had to be accomplished: a) allow the robot to play a sequence of notes or melody fluently; b) allow the robot to play notes accurately; c) allow the robot to adapt to multiple songs easily; d) allow the robot to be able to have social communication with participants; e) allow the robot to be able to deliver learning and therapy experiences to participants; and f) allow the robot to have fast responses and accurate decision-making. To accomplish these tasks, a module-based acoustic music interactive system was designed. Three modules were built in this intelligent system: Module 1: eye-hand self-calibration micro-adjustment; Module 2: joint trajectory generator; and Module 3: real-time performance scoring feedback (see Figure \\ref{module}).\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.9\\linewidth]{.\/fig\/module_blocks.eps}\\label{module}\\\\\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Block diagram of Module-based acoustic music interactive system.} \\label{module}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsection{Module 1: Color-Based Position Initiative}\n\tKnowledge about the parameters of the robot's kinematic model was essential for programming tasks requiring high precision, such as playing the Xylophone. While the kinematic structure was known because of the construction plan, errors could occur because of factors such as imperfect manufacturing. After multiple rounds of testing, it was determined that the targeted angle chain of arms never equals the returned chain. Therefore, we used a calibration method to eliminate this error.\n\t\n\tTo play the Xylophone, the robot had to be able to adjust its motions according to the estimated relative position of the Xylophone and the heads of the mallets it was holding. To estimate the poses adopted in this paper, we used a color-based technique.\n\t\n\tThe main idea in object tracking is that, based on the RGB color of the center blue bar, given a hypothesis about the Xylophone's position, one can project the contour of the Xylophone's model into the camera image and compare them to an observed contour. In this way, it is possible to estimate the likelihood of the position hypothesis. Using this method, the robot can track the Xylophone with extremely low cost in real-time (see Figure \\ref{color_detection}).\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.4\\linewidth]{.\/fig\/blue.eps}\\label{single_color_a}\\\\\n\t\t\t\t(a)\\\\\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.4\\linewidth]{.\/fig\/all_color.eps}\\label{single_color_b}\\\\\n\t\t\t\t(b)\\\\\n\t\t\t\n\t\t\t\t\\includegraphics[width=0.75\\linewidth]{.\/fig\/color_detection.eps}\\label{color_detection_c}\\\\\n\t\t\t\t(c)\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Color detection from NAO's bottom camera: (a) single blue color detection (b) full instrument color detection (c) color based edge detection.} \\label{color_detection}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\\subsection{Module 2: Joint Trajectory Generator}\n\tOur system parsed a list of hexadecimal numbers (from 1 to b) to obtain the sequence of notes to play. The system converted the notes into a joint trajectory using the striking configurations obtained from inverse kinematics as control points. The timestamps for the control points are defined by the user to meet the experiment requirement. The trajectory was then computed by the manufacturer-provided API, using Bezier curve \\cite{han2008novel} interpolation in the joint space, and then sent to the robot controller for execution. This process allowed the robot to play in-time with songs. \n\t\n\t\\subsection{Module 3: Real-Time Performance Scoring Feedback}\n\tTwo core features were designed to complete the task in the proposed scoring system: 1) music detection and 2) intelligent scoring feedback. These two functions could provide a real-life interaction experience using a music therapy scenario to teach participants social skills.\n\t\n\t\\subsubsection{Music Detection}\n\tMusic, from a science and technology perspective, is a combination of time and frequency. To make the robot detect a sequence of frequencies, we adopted the Short-time Fourier transform (STFT) for its audio feedback system. Doing so allowed the robot to be able to understand the music played by users and provide proper feedback as a music instructor.\n\t\n\tThe STFT is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then separately compute the Fourier transform for each shorter segment. Doing so reveals the Fourier spectrum on each shorter segment. The changing spectra can then be plotted as a function of time. In the case of discrete-time, data to be transformed can be broken up into chunks of frames that usually overlap each other to reduce artifacts at boundaries. Each chunk is Fourier transformed. The complex results are then added to a matrix that records magnitude and phase for each point in time and frequency (see Figure \\ref{stft}). This can be expressed as:\n\t\\begin{equation}\n\t\t\\resizebox{.6\\hsize}{!}{${\\displaystyle \\mathbf {STFT} \\{x[n]\\}(m,\\omega )\\equiv X(m,\\omega )=\\sum _{n=-\\infty }^{\\infty }x[n]w[n-m]e^{-j\\omega n}}$}\n\t\\end{equation}\n\t\\noindent likewise, with signal x[n] and window w[n]. In this case, m is discrete and $\\omega$ is continuous, but in most applications, the STFT is performed on a computer using the Fast Fourier Transform, so both variables are discrete and quantized. The magnitude squared of the STFT yields the spectrogram representation of the Power Spectral Density of the function:\n\t\\begin{equation}\n\t\t\\resizebox{.4\\hsize}{!}{${\\displaystyle \\operatorname {spectrogram} \\{x(t)\\}(\\tau ,\\omega )\\equiv |X(\\tau ,\\omega )|^{2}}$}\n\t\\end{equation}\n\t\\noindent After the robot detects the notes from user input, a list of hexadecimal numbers are returned. This list is used for two purposes: 1) to compare with the target list for scoring and sending feedback to the user and 2) to create a new input for having robot playback in the game session.\n\t\n\t\\begin{figure}\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/stft.eps}\\label{stft}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Melody detection with Short Time Fourier Transform} \\label{stft}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\subsubsection{Intelligent Scoring-Feedback System}\n\tTo compare the detected and target notes, we used Levenshtein distance, an algorithm that is typically used in information theory linguistics. This algorithm is a string metric for measuring the difference between two sequences.\n\t\n\tIn our case, the Levenshtein distance between two string-like hex-decimal numbers \n\t${\\displaystyle a,b}$ (of length ${\\displaystyle |a|}$ and ${\\displaystyle |b|}$ respectively) \n\tis given by ${\\displaystyle \\operatorname {lev} _{a,b}(|a|,|b|)}$,\n\t\\begin{equation}\n\t\t\\resizebox{.8\\hsize}{!}{${\\displaystyle \\qquad \\operatorname {lev} _{a,b}(i,j)={\\begin{cases}\\max(i,j)&{\\text{ if }}\\min(i,j)=0,\\\\\\min {\\begin{cases}\\operatorname {lev} _{a,b}(i-1,j)+1\\\\\\operatorname {lev} _{a,b}(i,j-1)+1\\\\\\operatorname {lev} _{a,b}(i-1,j-1)+1_{(a_{i}\\neq b_{j})}\\end{cases}}&{\\text{ otherwise.}}\\end{cases}}}$}\\\\\n\t\\end{equation}\n\t\\noindent where ${\\displaystyle 1_{(a_{i}\\neq b_{j})}}$ is the indicator function equal to 0 when \n\t${\\displaystyle a_{i}=b_{j}}$ and equal to 1 otherwise, and ${\\displaystyle \\operatorname {lev} _{a,b}(i,j)}$ \n\tis the distance between the first ${\\displaystyle i}$ characters of ${\\displaystyle a}$ and the\n\tfirst ${\\displaystyle j}$ characters of ${\\displaystyle b}$.\n\tNote that the first element in the minimum corresponds to deletion (from ${\\displaystyle a}$ to \n\t${\\displaystyle b}$), the second to insertion and the third to match or mismatch, depending on \n\twhether the respective symbols are the same. \n\t\n\tBased on the real-life situation, we defined a likelihood margin for determining whether the result\n\tis good or bad:\n\t\\begin{equation}\n\t\t\\resizebox{.45\\hsize}{!}{${likelihood = \\dfrac{len(target) - lev_{target,source}}{len(target)}}$}\n\t\\end{equation}\n\t\\noindent where, if the likelihood is more than 66\\% (not including single note practice since, in that case, it is only correct or incorrect), the system will consider it to be a good result. This result is then passed to the accuracy calculation system to have the robot decide whether it needs to add additional practice trials (e.g., 6 correct out of a total of 10 trials).\n\t\n\t\\section{Social Behavior Results}\n\t\\subsection{Motor Control}\n\tNine ASD and seven TD participants completed the study over the course of eight months. All ASD participants completed all six sessions (baseline, intervention, and exit) while all the TD subjects completed the required baseline and exit sessions. Conducting a Wizard of Oz experiment, a well-trained researcher was involved in the baseline and exit sessions in order for there to be high-quality observations and performance evaluations. With well-designed, fully automated intervention sessions, NAO was able to initiate music-based therapy activities with participants.\n\t\n\tSince the music detection method was sensitive to the audio input, a clear and long-lasting sound from the Xylophone was required. As seen in Figure \\ref{fig9}, it is evident that most of the children were able to strike the Xylophone properly after one or two sessions (the average accuracy is improved over sessions). Notice that participants 101 and 102 had significant improvement during intervention sessions. Some of the participants started at a higher accuracy rate and sustained a rate above 80\\% for the duration of the study. Even with some oscillation, participants with this type of accuracy rate were considered to have consistent motor control performance. \n\t\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure9.eps}\\label{fig9}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Motor control accuracy result.} \\label{fig9}\n\t\t\\end{center}\n\t\\end{figure}\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure10.eps}\\label{fig10}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Main music therapy performance accuracy.} \\label{fig10}\n\t\t\\end{center}\n\t\\end{figure}\n\tFigure \\ref{fig10} shows the accuracy of the first music therapy activity that was part of the intervention sessions across all participants. As described in the previous section, the difficulty level of this activity was designed to increase across sessions. With this reasoning, the accuracy of participant performance was expected to decrease or remain consistent. This activity required participants to be able to concentrate, use joint attention skills during the robot therapy stage, and respond properly afterwards.  As seen in Figure \\ref{fig10}, most of the participants were able to complete single\/multiple notes practice with an average of a 77.36\\%\/69.38\\% accuracy rate although, even with color hints, the pitch difference of notes was still a primary challenge. Due to the difficulty of sessions 4 and 5, a worse performance, when compared to the previous two sessions, was acceptable. However, more than half of the participants consistently showed a high accuracy performance or an improved performance than previous sessions. \n\t\n\t\\subsection{Turn-Taking Behavior}\n    The American Psychological Association \\cite{vandenbos2007american} defines turn-taking as, \"in social interactions, alternating behavior between two or more individuals, such as the exchange of speaking turns between people in conversation or the back-and-forth grooming behavior that occurs among some nonhuman animals. Basic turn-taking skills are essential for effective communication and good interpersonal relations, and their development may be a focus of clinical intervention for children with certain disorders (e.g., Autism).\" Learning how to play one's favorite song can be a motivation that helps ASD participants understand and learn turn-taking skills. In our evaluation system, the annotators scored turn-taking based on how well kids were able to follow directions and display appropriate behavior during music-play interaction. In other words, participants needed to listen to the robot's instructions and then play, and vice versa. Any actions not following the interaction routine were considered as less or incomplete turn-taking. Measurements are described as follows: measuring turn-taking behavior can be subjective, so to quantify this, a grading system was designed. Four different behaviors were defined in the grading system: (a) \"well-done\", this level is considered good behavior where the participant should be able to finish listening to the instructions from NAO, start playback after receiving the command, and wait for the result without interrupting (3 points for each \"well-done\"); (b) \"light-interrupt\", in this level, the participant may exhibit slight impatience, such as not waiting for the proper moment to play or not paying attention to the result (2 points given in this level); (c) \"heavy-interrupt\", more interruptions may accrue in this level and the participant may interrupt the conversation, but is still willing to play back to the robot (1 point for this level); and (d) \"indifferent\", participant shows little interest in music activities including, but not limited to, not following behaviors, not being willing to play, not listening to the robot or playing irrelevant music or notes (score of 0 points). The higher the score, the better turn-taking behavior the participant demonstrated. All scores were normalized into percentages due to the difference of total \"conversation\" numbers. Figure \\ref{fig11} shows the total results among all participants. Combining the report from video annotators, 6 out of 9 participants exhibited stable, positive behavior when playing music, especially after the first few sessions. Improved learn-and-play turn-taking rotation was demonstrated over time. Three participants demonstrated a significant increase in performance, suggesting turn-taking skills were taught in this activity. Note that two participants (103 \\& 107) had a difficult time playing the Xylophone and following turn-taking cues given by the robot.\n\t\n\t\\begin{figure}[tbp]\n\t\t\\begin{center}\n\t\t\t\\begin{center}\n\t\t\t\n\t\t\t\t\\includegraphics[width=1\\linewidth]{.\/fig\/figure11.eps}\\label{fig11}\\\\\n\t\t\t\\end{center}\n\t\t\t\\caption{Normalized turn-taking behavior result for all subjects during intervention sessions.\n\t\t\t} \\label{fig11}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\\section{Music Emotion Classification Results}\n\t\\ Since we developed our emotion classification method based on the time-frequency analysis of EDA signals, the main properties of The Continuous Wavelet Transforms (CWT), assuming that it is a C-Morlet wavelet, is presented here. Then, the pre-processing steps, as well as the wavelet-based feature extraction scheme, are discussed. Finally, we briefly review the characteristics of the SVM, the classifier used in our approach.\n\t\n\tEDA signals were collected in this study. By using the annotation and analysis method from our previous study \\cite{feng2018wavelet}, we were able to produce a music-event-based emotion classification result that is presented below. To determine the emotions of the ASD group, multiple comparisons were made after annotating the videos. Note that on average, 21\\% of emotions were not clear during the annotation stage. It is necessary to have them as unclear rather than label them with specific categories.\n\t\n\tDuring the first part of the annotation, it was not obvious to conclude the facial expression changes from different activities, however, in S1 participants showed more \"calm\" for most of the time as the dominant emotion due to the simplicity of completing the task. In S2, the annotator could not have a precise conclusion by reading the facial expression from participants as well. Most of the participants intended to play music and complete the task, however, feelings of \"frustration\" were often invoked in this group. A very similar feeling can be found in S3 when compared to S1. In S3, all the activities were designed to create a role changing environment for users in order to stimulate a different emotion. Most of the children showed \"happy\" emotion during the music game section.\n\t We noticed that different activities can result in a change in emotional arousal. As mentioned above, the warm-up section and single activity practice section use the same activity with different intensities levels. The gameplay section has the lowest difficulty and is purposely designed to be more relaxing.\n\t\\begin{table*}\n\t\t\\label{tab1}\n\t\t\\begin{center}\n\t\t\t\\caption{Emotion change in different events using wavelet-based feature extraction under SVM classifier. }\n\t\t\t\\vspace{3mm}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t& Kernels                     & Accuracy(\\%) & AUC & Precision(\\%) & Recall(\\%) \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{Linear}}     & 75       & .78  & 76        & 72     \\\\\n\t\t\t\tS1 vs S3       &                             & 57       & .59  & 56        & 69     \\\\\n\t\t\t\tS2 vs S3       &                             & 69       & .72  & 64        & 86     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{52}              \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{Polynomial}} & 66       & .70  & 70        & 54     \\\\\n\t\t\t\tS1 vs S3       &                             & 64       & .66  & 62        & 68     \\\\\n\t\t\t\tS2 vs S3       &                             & 65       & .68  & 62        & 79     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{50}              \\\\\n\t\t\t\t\\hline\n\t\t\t\tS1 vs S2       & \\multirow{4}{*}{\\textbf{RBF}}        & 76       & .81  & 76        & 75     \\\\\n\t\t\t\tS1 vs S3       &                             & 57       & .62  & 57        & 69     \\\\\n\t\t\t\tS2 vs S3       &                             & 70       & .76  & 66        & 83     \\\\\n\t\t\t\tS1 vs S2 vs S3 &                             & \\multicolumn{4}{l}{53}              \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\t\\label{tab1}\n\t\t\\end{center}\n\t\\end{table*}\n\t\n\n\t\\begin{table*}[tbp]\n\t\t\\label{tab2}\n\t\t\\begin{center}\n\t\t\t\\caption{Emotion change classification performance in single event with segmentation using both SVM and KNN classifier. }\n\t\t\t\\vspace{3mm}\n\t\t\t\\resizebox{\\columnwidth}{!}{\n\t\t\t\t\\begin{tabular}{ccccccccc}\n\t\t\t\t\t\\multicolumn{1}{l}{\\multirow{3}{*}{}} & \\multicolumn{5}{c}{Segmentation Comparison in Single Task}                                                                                                       \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multicolumn{1}{l}{}                  & \\multicolumn{4}{c}{Warm up Section}                                                   & \\multicolumn{4}{c}{Song Practice Section}                                                  \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multicolumn{1}{l}{}                  & Kernels                     & Accuracy (\\%) & K value                & Accuracy(\\%) & Kernels                     & Accuracy(\\%) & K value                & Accuracy(\\%) \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{Linar}      & 52.62    & \\multirow{4}{*}{K = 1} & 54       & \\multirow{4}{*}{Linar}      & 53.79    & \\multirow{4}{*}{K = 1} & 52.41    \\\\\n\t\t\t\t\tlearn vs feedback                                   &                             & 53.38    &                        & 50.13    &                             & 53.1     &                        & 51.72    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 47.5     &                        & 50.38    &                             & 54.31    &                        & 50.86    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 35.08    &                        & 36.25    &                             & 35.52    &                        & 36.55    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{Polynomial} & 49       & \\multirow{4}{*}{K = 3} & 50.25    & \\multirow{4}{*}{Polynomial} & 53.79    & \\multirow{4}{*}{K = 3} & 50.69    \\\\\n\t\t\t\t\tlearn vs feedback                                 &                             & 50.75    &                        & 50.13    &                             & 50.86    &                        & 50.34    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 49.87    &                        & 49.5     &                             & 49.14    &                        & 52.07    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 33.92    &                        & 35.83    &                             & 34.71    &                        & 35.29    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tlearn vs play                                   & \\multirow{4}{*}{RBF}        & 54.38    & \\multirow{4}{*}{K = 5} & 48.37    & \\multirow{4}{*}{RBF}        & 50.86    & \\multirow{4}{*}{K = 5} & 50.17    \\\\\n\t\t\t\t\tlearn vs feedback                                   &                             & 55.75    &                        & 52.75    &                             & 53.97    &                        & 50.17    \\\\\n\t\t\t\t\tplay vs feedback                                   &                             & 51.12    &                        & 50       &                             & 53.79    &                        & 52.93    \\\\\n\t\t\t\t\tlearn vs play vs feedback                                 &                             & 36.83    &                        & 34.17    &                             & 34.83    &                        & 33.1    \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\n\t\t\t}\n\t\t\t\\label{tab2}\n\t\t\\end{center}\n\t\n\t\\end{table*}\n\t\\begin{table*}[]\n\t\t\n\t\t\\begin{center}\n\t\t\t\\caption{Classification rate in children learn, children play and robot feedback across\n\t\t\t\twarm up (S1) and music practice (S2) sessions.}\n\t\t\t\\vspace{3mm}\n\t\t\t\\label{tab3}\n\t\t\t\\begin{tabular}{lcccccc}\n\t\t\t\t\\multicolumn{1}{c}{\\multirow{2}{*}{}} & \\multicolumn{3}{c}{Accuracy of SVM} & \\multicolumn{3}{c}{Accuracy of KNN} \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multicolumn{1}{c}{}                  & Linear   & Polynomial    & RBF    & K = 1   & K = 3     & K = 5     \\\\\n\t\t\t\t\\hline\n\t\t\t\tlearn 1 vs learn 2                                 & 73.45    & 69.31   & 80.86  & 73.28   & 71.03   & 65      \\\\\n\t\t\t\t\\hline\n\t\t\t\tplay 1 vs play 2                                 & 75.34    & 68.79   & 80     & 74.48   & 69.14   & 64.31   \\\\\n\t\t\t\t\\hline\n\t\t\t\tfeedback 1 vs feedback 2                                 & 76.38    & 69.48   & 80.34  & 74.14   & 69.14   & 66.9   \n\t\t\t\\end{tabular}\n\t\t\t\n\t\t\t\\label{tab3}\n\t\t\\end{center}\n\t\\end{table*}\n\t\n\t\\begin{table}[]\n\t\t\\label{tab4}\n\t\t\n\t\t\\begin{center}\n\t\t\t\\caption{TD vs ASD Emotion Changes from Baseline and Exit Sessions.}\n\t\t\t\\vspace{3mm}\n\t\t\t\\begin{tabular}{llllll}\n\t\t\t\t& \\textbf{Linear} & \\textbf{Polynomial} & \\textbf{RBF} \\\\\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\\textbf{Accuracy}                          & 75              & 62.5                & 80           \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\multirow{2}{*}{\\textbf{Confusion Matrix}} & 63  37          & 50  50              & 81  19       \\\\\n\t\t\t\t& 12  88          & 25  75              & 25  75       \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\t\\label{tab4}\n\t\t\\end{center}\n\t\\end{table}\n\t\n\tIn the first part of the analysis, EDA signals were segmented into small event-based pieces according to the number of \"conversations\" in each section. One \"conversation\" was defined by three movements: 1) robot\/participant demonstrates the note(s) to play; 2) robot\/participant repeats the note(s); and 3) robot\/participant presents the result. Each segmentation lasts about 45 seconds. The CWT of the data, assuming the use of the C-Morlet wavelet function, was used inside a frequency range of (0.5, 50)Hz. A Support Vector Machines (SVM) classifier was then employed to classify \"conversation\" segmentation among three sections using the wavelet-based features. Table \\ref{tab1}shows the classification accuracy for the SVM classifier with different kernel functions. As seen in this table, emotion arousal change between warmup (S1) and music practice (S2) and S2 and music game (S3) sections can be classified using a wavelet-based feature extraction SVM classifier with an average accuracy of 76\\% and 70\\%, respectively. With the highest percentage of accuracy for S1 and S3 being 64\\%, fewer emotion changes between the S1 and S3 sections may be indicated. \n\t\n\tIn the second part of the analysis, EDA signals were segmented into small event-based pieces according to the number of \"conversations\" in each section as mentioned previously. In order to discover the emotion fluctuation inside one task, each \"conversation\" section was carefully divided into 3 segments, as described before. Again, each segmentation lasts about 45 seconds.\n\t Each segment lasts about 10 - 20 seconds. Table \\ref{tab2} shows the full result of emotion fluctuation in the warm-up (S1) and music practice (S2) sections from the intervention session. Notice that all of the segments cannot be appropriately classified using the existing method. Both SVM and KNN show stable results. This may suggest that the ASD group has less emotion fluctuation or arousal change once the task starts, despite varying activities in it. Stable emotion arousal in a single task could also benefit from the proper activity content, including robot agents playing music and language used during the conversation. Friendly voice feedback was based on the performance delivered to participants who were well prepared and stored in memory. Favorable feedback occurred while receiving correct input and in the case of incorrect playing, participants were given encouragement. Since emotion fluctuation can affect learning progress, less arousal change indicates the design of intervention sessions that are robust. \n\t\n\tCross-section comparison is also presented below. Since each \"conversation\" contains 3 segments, it is necessary to have specific segments from one task to compare with the other task it corresponded to. Table \\ref{tab3} shows the classification rate in children learn, children play, and robot feedback across warm-up (S1) and music practice (S2) sessions. By using RBF kernel, wavelet-based SVM classification rate had an accuracy of ~80\\% for all 3 comparisons. This result also matches the result from Table \\ref{tab1}. \n\t\n\tThe types of activities and procedures between the baseline and exit session for both groups were the same. Using the \"conversation\" concept above, each were segmented. Comparing with target and control groups using the same classifier, an accuracy rate of 80\\% for detecting different groups was found (see Table \\ref{tab3}). Video annotators also reported \"unclear\" in reading facial expressions from the ASD group. Taken together, these results suggest that even with the same activities, TD and ASD groups display different bio-reactions. It has also been reported that a significant improvement of music performance was shown in the ASD group (see Table \\ref{tab4}), although both groups had similar performance at their baseline sessions. Furthermore, the TD group was found to be more motivated to improve and ultimately perfect their performance, even when they made mistakes.\n\t\n\tWhen comparing emotion patterns from baseline and exit sessions between TD and ASD groups in Table \\ref{tab3}, differences can be found. This may suggest that we have discovered a potential way of using biosignals to help diagnose autism at an early age. According to annotators and observers, TD participants showed a strong passion for this research. Excitement, stress, and disappointment were easy to recognize and label when watching the recorded videos. On the other hand, limited facial expression changes were detected in the ASD group. This makes it challenging to determine whether the ASD participants had different feelings or had the same feelings but different biosignal activities compared to the TD group.\n\t\n\t\\section{Discussion, Conclusion and Future Work}\n\t\\ As shown by others \\cite{lagasse2019assessing,lim2011effects} as well as this study, playing music to children with ASD in therapy sessions has a positive impact on improving their social communication skills. Compared to the state of the art \\cite{dvir2020body,bhat2013review, corbett2008brief}, where they utilized both recorded and live music in interventional sessions for single and multiple participants, our proposed robot-based music-therapy platform is a promising intervention tool in improving social behaviors, such as motor control and turn-taking skills. As our human robot interaction studies (over 200 sessions were conducted with children), most of the participants were able to complete motor control tasks with ~70\\% accuracy and 6 out of 9 participants demonstrated stable turn-taking behavior when playing music. The emotion classification SVM classifier presented illustrated that emotion arousal from the ASD group can be detected and well recognized via EDA signals. \n\t\n\tThe automated music detection system created a self-adjusting environment for participants in early sessions. Most of the ASD participants began to develop the strike movement in the initial two intervention sessions; some even mastered the motor ability throughout the very first warm-up event. The robot was able to provide verbal directions and demonstrations to participants by providing voice command input as applicable, however, the majority of the participants did not request this feedback, and instead just focused on playing with NAO. This finding suggests that the young ASD population can learn fine motor control ability from specific, well-designed activities.\n\t\n\tThe purpose of using a music therapy scenario as the main activity in the current research was to create an opportunity to practice a natural turn-taking behavior during social interaction. Observing all experimental sessions, six out of nine participants exhibited proper turn-taking after one or two sessions, suggesting the practice helped improve this behavior. Specifically, participant 107 significantly improved in the last few sessions when comparing results of the baseline and final session. Participant 109 had trouble focusing on listening to the robot most of the time, however, with prompting from the researcher, they performed better at the music turn-taking activity for a short period of time. For practicing turn-taking skills, fun, motivating activities, such as the described music therapy sessions incorporating individual song preference, should be designed for children with autism.\n\t\n\tDuring the latter half of the sessions, participants started to recognize their favorite songs. Even though the difficulty for playing proper notes was much higher, over half of the participants became more focused in the activities. Upon observations, it became clear that older participants spent more time interacting with the activities during the song practice session compared to younger participants, especially during the half\/whole song play sessions. This could be for several reasons. First, the more complex the music, the more challenging it is, and the more concentration participants need to be successful. Older individuals may be more willing to accept the challenge and are better able to enjoy the sense of accomplishment they receive from their verbal feedback at the end of each session. Prior music knowledge could also be another reason for this result as older participants may have had more opportunities to learn music at school. \n\t\n\tThe game section of each session reflected the highest interest level, not only because it was relaxing and fun to play, but also because it was an opportunity for the participants to challenge the robot to mirror their free play. This exciting phenomenon could be seen as a game of \"revenge.\" Participant 106 exhibited this behavior by spending a significant amount of time in free play game mode. According to the session executioner and video annotators, this participant, the only girl, showed very high level of involvement for all the activities, including free play. Based on the conversation and music performance with the robot, participant 106 showed a strong interest in challenging the robot in a friendly way. High levels of engagement further supports the idea that the proposed robot-based music-therapy platform is a viable interventional tool. Additionally, these findings also highlight the need for fun and motivating activities, such as music games, to be incorporated into interventions designed for children with autism.\n\t\n\tConducting emotion studies with children with autism can be difficult and bio-signals provide a possible way to work around the unique challenges of this population. The event-based emotion classification method adopted in this research suggests that not only are physiological signals a good tool to detect emotion in ASD populations, but it is also possible to recognize and categorize emotions using this technique. The same activity with different intensities can cause emotion change in the arousal dimension, although, for the ASD group, it is difficult to label emotions based on facial expression changes in the video annotation phase. Fewer emotion fluctuations in a particular activity, as seen in Table \\ref{tab2}, suggests that a mild, friendly, game-like therapy system may encourage better social content learning for children with autism, even when there are repetitive movements. These well-designed activities could provide a relaxed learning environment that helps participants focus on learning music content with proper communication behaviors. This may explain the improvement in music play performance during the song practice (S2) section of intervention sessions, as seen in Figure \\ref{fig10}.\n\t\n\tThere were several limitations of this study that should be considered. As is the case with most research studies, a larger sample size is needed to better understand the impact of this therapy on children with ASD. The ASD group in this paper also only included one female. Future work should include more females to better portray and understand an already underrepresented portion of this population of interest. In addition, improving skills as complex as turn-taking for example, likely requires more sessions over a longer period of time to better enhance the treatment and resulting behavioral changes. Music practice can also be tedious if it is the only activity in an existing interaction system and may have influenced the degree of interaction in which participants engaged with the robot. Future research could include more activities or multiple instruments, as opposed to just one, to further diversify sessions. The choice of instrument, and its resulting limitations, may also have an impact. The Xylophone, for example, is somewhat static and future research could modify or incorporate a different instrument that can produce more melodies, accommodate more complex songs, and portray a wider range of musical emotion and expression. Finally, another limitation of this study was the type of communication the robot and participant engaged in. Verbal communication was not rich between participants and NAO, and instead, most of the time, participants could follow the instructions from the robot without asking it for help. While this was likely not an issue for the specific non-verbal behavioral goals of this intervention, future work should expand upon this study by incorporating other elements of behavior that often have unique deficits in this population, including speech.\n\t\n\tIn summary, this paper presented a novel robot-based music-therapy platform to model and improve social behaviors in children with ASD. In addition to the novel platform introduced, this study also incorporated emotion recognition and classification utilizing EDA physiological signals of arousal. Results of this study are consistent with findings in the literature for TD and ASD children and suggest that the proposed platform is a viable tool to facilitate the improvement of fine motor and turn-taking skills in children with ASD.\n\t\n\t\\section*{Acknowledgment}\n\t\\ This research was partially supported by gifts for research on autism to the University of Denver from several family members. The researchers would like to thank the children and their family members for their dedication and willingness to volunteer their time and make this research possible.\n\t\n\t\\bibliographystyle{frontiersinSCNS_ENG_HUMS}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nLet $n \\geq 1$ and let $P_n$ be a directed path on $[n] = \\{1,2,\\ldots,n\\}$ with directed edges from $i$ to $i-1$ for $i=2,3,\\ldots,n.$ Let $1 \\leq m \\leq n$ and assume that $m$ drivers arrive at vertex $n$ one by one, with the $i$th driver willing to park in vertex $X_i \\in [n].$ If the $i$th driver finds $X_i$ empty, they park there. If not, they continue their drive towards $1,$ parking in the first available parking space. If no such spot can be found, the driver leaves the path without parking. We say that $(x_1, \\ldots, x_m),$ with $x_1, \\ldots, x_m \\in [n],$ is a \\em parking function \\em for $P_n$ if for $X_i = x_i$ for $1 \\leq i \\leq m,$ all $m$ drivers park on the path.\n\nParking functions were first studied in the 1960s by Konheim and Weiss \\cite{KonheimWeiss-parkingPath}. They evaluated the number of parking functions, which is equivalent to evaluating the probability that an $m$-tuple of independent random variables uniformly distributed on $[n]$ gives a parking function. A similar model, with $P_n$ replaced by a uniform random rooted Cayley tree on $[n]$ was studied by Lackner and Panholzer \\cite{LacknerPanholzer}. Motivated by finding a probabilistic explanation for some phenomenons observed in \\cite{LacknerPanholzer}, Goldschmidt and Przykucki \\cite{GoldschmidtPrzykucki} analyzed the parking processes on critical Galton-Watson trees, as well as on trees with Poisson(1) offspring distribution conditioned on non-extinction, in both cases with the edges directed towards the root. Note that in all the setups above, drivers have only one choice of route at any time of the process.\n\nIn this paper, we are concerned with a related model, introduced by Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS}, in which the cars move at random. Let $L=(V,E)$ be a Cayley graph on a group $V$ with generating set $R$, and let  $\\mu$ be a probability distribution on $R$: we will refer to such a triple $(L,R,\\mu)$ as a {\\em car park triple}. At time $0$, each position $v \\in V$ is independently assigned a car with probability $p$ \nor a parking space with probability $1-p$. The cars follow independent random walks with increments  $\\mu$ \nand each car continues to follow the random walk until it finds a free space where it parks (if more than one car arrives at a free space at the same time, then one is chosen to park according to some rule).\\footnote{We note that Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS} work in a slightly more general setting, see Section 2 in \\cite{DGJLS}. While our results in Section \\ref{sec:definitions} also hold in the setting in \\cite{DGJLS}, we believe that the class of Cayley graphs is a fairly general setting and the link with lattices is a little clearer.}\n\nWe are interested in the distribution of journey lengths of cars. We introduce the stopping time $\\tau^v$ where $\\tau^v = 0$ if position $v$ is a parking space, and otherwise $\\tau^v$ is the time the car starting at $v$ takes to park ($\\tau^v= \\infty$ if the car never parks). We also write $\\tau = \\tau^0$ (by symmetry we only need to consider $v=0$). Given $t \\ge 0$ and a vertex $v,$ let\n\\[\nV^v(t)= \\left|\\left\\{(u,s) \\in V \\times [t] : \\mbox{car $u$ visits $v$ at time $s$}\\right\\}\\right| + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}}\n\\]\nbe the number of cars that visit $v$ up to time $t.$\n\nIn the particular case of the lattice ${\\mathbb Z}^d$ (with edges joining lattice points at Euclidean distance 1), Damron, Gravner, Junge, Lyu, and Sivakoff \\cite{DGJLS} prove the following theorem.\n\\begin{thm}\\label{thm:DGJLS}\nConsider the parking process on $\\mathbb{Z}^d$ with simple symmetric random walks.\n\\begin{enumerate}\n \\item If $p \\geq 1\/2$ then  $\\mathbb{E}[\\tau] = \\infty$ with $\\mathbb{E}[\\min\\{\\tau,t\\}] = (2p-1)t + o(t).$\n \\item If $p < 1\/2$ then $\\tau$ is almost surely finite. Moreover, if $p < (256d^6e^2)^{-1}$ then $\\mathbb{E}[\\tau] < \\infty.$\n\\end{enumerate}\n\\end{thm}\n\nFor $p > 1\/2,$ Theorem \\ref{thm:DGJLS} gives good asymptotics for $\\mathbb{E}[\\min\\{\\tau,t\\}].$ However, \nfor $p = 1\/2$ Theorem \\ref{thm:DGJLS} only tells us that $\\mathbb{E}[\\min\\{\\tau,t\\}]$ is $o(t),$ while the authors of \\cite{DGJLS} conjecture that for $d=1$ and $p=1\/2$ we have $\\mathbb{E}[\\min\\{\\tau,t\\}] = \\Theta(t^{3\/4})$ \\cite{Junge-seminar}. \nMoreover, \nfor $d=1$,  Theorem \\ref{thm:DGJLS} only gives $\\mathbb{E}[\\tau] < \\infty$ for $p < 0.000528,$ while it is conjectured that this holds for all $p < 1\/2.$\n\nHere, we address both conjectures when $d=1,$ and prove the following two theorems.\n\n\\begin{thm}\\label{thm:p=1\/2}\nFor the parking problem on $\\mathbb{Z},$ when $p=1\/2,$ there exist constants $C, c > 0$ such that\n\\[\n c t^{3\/4} (\\log t)^{-1\/4} \\leq \\mathbb{E}[\\min\\{\\tau,t\\}] \\leq C t^{3\/4}.\n\\]\n\\end{thm}\n\n\\begin{thm}\\label{thm:p<1\/2}\nFor the parking problem on $\\mathbb{Z},$ when $p<1\/2$ we have $\\mathbb{E}[\\tau] < \\infty.$\n\\end{thm}\n\nFor the parking process on $\\mathbb{Z}$ with $p=1\/2,$ Theorem \\ref{thm:p=1\/2} gives good bounds on the asymptotic growth of $\\mathbb{E}[\\min\\{\\tau,t\\}]$ by showing that it indeed equals $t^{3\/4}$ up to a fractional power of $\\log t.$ For $p < 1\/2,$ Theorem \\ref{thm:p<1\/2} confirms that the expected journey length of a car is finite as predicted. Damron, Gravner, Junge, Lyu and Sivakoff \\cite{DGJLS} also ask whether (for a large family of parking processes)  there is a critical exponent $\\gamma >0$ such that, for some constant $C > 0$, $\\mathbb{E}[\\tau] \\sim C\\left(1\/2 -p\\right)^{-\\gamma}$ as $p$ increases to $1\/2.$ For the parking problem on $\\mathbb{Z}$, we have a partial result in this direction (though we have no conjecture as to the value, or even the existence of a critical exponent).\n\n\\begin{thm}\\label{thm:critexp}\nFor the parking problem on $\\mathbb{Z},$ $\\mathbb{E}[\\tau] = O\\left((1\/2 - p)^{-6}\\right)$ as $p\\nearrow1\/2.$\n\\end{thm}\n\nIn this paper, we will consider strategies that modify the car-parking process. \nWe will introduce two types of strategy: parking strategies where we allow cars to choose whether or not to park in an available space, and car removal strategies where we remove cars from the parking process (we defer formal definitions to Section \\ref{sec:definitions}).\n For a  car park triple $(L,R,\\mu)$ and a strategy $S$, we will write $V^v_S(t)$ for the value of $V^v(t)$ when strategy $S$ is followed, and similarly $\\tau^v_S$; we write $G$ for the greedy strategy (i.e.~the original process).\n\n\nThe key properties of parking and car removal strategies that we shall use are given in the following theorems, which show that no parking strategy is quicker than the greedy one, and that adding car removal makes parking easier. \nWe note that these results hold in the more general setting of Cayley graphs.\n\n\\begin{thm}\\label{thm:supermarket}\nLet $S$ be a parking strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0$ and vertices $v$, the distribution of $V^v_G(t)$ majorises $V^v_S(t)$. That is for all $t, k \\ge 0$,\n\\[\n\\mathbb{P}\\left[V^v_G(t) \\le k\\right] \\ge \\mathbb{P}\\left[V^v_S(t) \\le k\\right].\n\\]\n\\end{thm}\n\n\\begin{thm}\n\\label{thm:pinkfloyd}\nLet $Q$ be a car removal strategy on the car park triple $(L,R,\\mu)$. Then $\\tau^v_Q \\le \\tau^v_G$, and for all $t \\ge 0$ we have $V^v_Q(t) \\le V^v_G(t)$.\n\\end{thm}\n\nIn order to prove Theorem \\ref{thm:supermarket} we introduce a new model, in which the cars follow directions stored at the vertices they visit, rather than their own individual random walks.  We will refer to this as the {\\em space-based model}, in contrast to the {\\em car-based model} described above.  Even though the stochastic properties of the two models are equivalent, the new model allows us to control the quantity $V^v(t)$ better, and we are then able to easily deduce the desired result for the original parking problem.\n\nThe paper is organised as follows. In Section \\ref{sec:definitions} we define the parking processes, introduce the notions of parking strategies and car removal, and prove Theorems \\ref{thm:supermarket} and \\ref{thm:pinkfloyd}. This allows us to consider both more and less restrictive parking problems, which we use in our arguments. In Section \\ref{sec:probabilities} we recall some known probability bounds that are used in this paper. In Section \\ref{sec:p=1\/2upper} we prove the upper bound on $\\mathbb{E}[\\min\\{\\tau,t\\}]$ in Theorem \\ref{thm:p=1\/2}, and in Section \\ref{sec:p=1\/2lower} we prove the lower bound. In Section \\ref{sec:p<1\/2} we prove Theorems \\ref{thm:p<1\/2}. and \\ref{thm:critexp}. Finally in Section \\ref{yaymorewaffle} we conclude the paper with some related problems and open questions.\n\nThroughout this paper, we use the notation $a \\wedge b = \\min\\{a,b\\}.$ For a normally distributed random variable $Z$ with mean $0$ and variance $1,$ we write $\\Phi(x) = \\mathbb{P}\\left[Z \\le x\\right].$\n\n\\section{Model specifics, parking strategies, and car removal}\n\\label{sec:definitions}\n\nWe will want to consider slight modifications of the original parking problem on $\\mathbb{Z}.$ In this section, we introduce new notation for these modifications and also compare these modifications to the original problem. The first modification is the addition of \\emph{parking strategies}. The second is the addition of \\emph{car removal} to the process. We compare the expected journey length of a car by time $t,$ showing that non-trivial parking strategies increase expected journey times while car removal decreases them. In fact, we are able to show that these bounds hold for any car park triple.\n\n\\subsection{The car-based parking model}\n\nLet us recall some definitions.\nLet $H$ be a group and $R$ be a generating set for $H$. The \\em Cayley graph \\em of $H$ with respect to $R$ is the edge-coloured directed graph $L = (H,E)$ where $$E := \\{(h,hr) : h \\in H, r \\in R\\},$$ and the edge $(h,hr)$ is coloured $r$.\nNote that if $R$ is closed under taking inverses then $(x,y) \\in E$ if and only if $(y,x) \\in E,$ and so we can just consider the underlying graph. For example, the $d$-dimensional integer-lattice $\\mathbb{Z}^d$ can be thought of as the abelian group with generating set $\\{e_1,-e_1,\\ldots,e_d,-e_d\\} \\subset \\mathbb{Z}^d$ where the $i$-th co-ordinate of $e_i$ is $1$ and all others are $0$.\n\n\n\nA {\\em car park triple} is an ordered triple $(L,R,\\mu)$ , where\n$L=(V,E)$ is a Cayley graph on a group $V$ with generating set $R$ and $\\mu$ is a probability distribution on $R$ \n (In later sections we will be interested in the parking problem on $\\mathbb{Z}$, namely the triple $(\\mathbb{Z},\\{-1,+1\\},\\mu^{\\mathbb{Z}})$ where $\\mu^{\\mathbb{Z}}(-1) = \\mu^{\\mathbb{Z}}(+1) = 1\/2$.  However, the results in this section hold in the more geenral model.)\n\nWe define the parking problem on the car park triple $(L,R,\\mu)$ as follows.\n\n\\begin{defn}\\label{defn:free parking}\nIndependently for each vertex $v \\in V$, let: \n\\begin{itemize}\n\\item $X^v = (X^v_0,X^v_1,\\ldots)$ be a Markov chain on $L$ with $X^v_0=v$, and transition matrix $(p_{u,w})$ where $p_{u,ur} = \\mu(r)$ for each $u\\in V$ and $r\\in R$, and $p_{u,w} = 0$ otherwise.\n\n\\item $(U^v_s)_{s \\in \\mathbb{N}}$ be a sequence of independent $\\mathrm{Unif}([0,1])$ random variables. \n\n\\item $B^v$ be a $\\mbox{Bernoulli}(p)$ random variable. We initially place a car at $v$ when $B^v = 1$ and otherwise a parking space with the capacity for one car.\n\\end{itemize}\n\nA car starting at vertex $v$ moves according to the Markov chain $X^v$ until it finds a free parking space and parks there. (We do not use the random walks $X^v$ for those $v$ where we initially place a parking space; we define them just for the simplicity of the model.) If cars $v_1,\\ldots,v_k$ all arrive at the same free parking space at time $s,$ we park car $v_j$ with smallest $U^{v_j}_s.$\n\\end{defn}\n\nWe shall sometimes refer to the model in Definition \\ref{defn:free parking} as the \\emph{car-based parking model}.\n\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space. A \\emph{filtration} is a sequence $\\mathcal{F}_0 \\subseteq \\mathcal{F}_{1} \\subseteq  \\ldots$ of $\\sigma$-algebras. A random variable $\\tau : \\omega \\rightarrow \\mathbb{N}$ is a \\emph{stopping time} with respect to a filtration $(\\mathcal{F}_{t})^{\\infty}_{t=0}$ if $\\tau^{-1}(\\{t\\}) \\in \\mathcal{F}_{t}$ for each $t \\in \\mathbb{N}.$  In the car-based model, for the parking problem on the car park triple $(L,R,\\mu)$, the probability space $(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge0},\\mathbb{P})$ is as expected, with a filtration $\\left(\\mathcal{F}_t\\right)_{t \\ge 0}$ defined by\n\\[\n\\mathcal{F}_t = \\sigma((B^v)_{v \\in V}, (X^v_s)_{v \\in V,0 \\leq s \\leq t}, (U^v_s)_{v \\in V,1 \\leq s \\leq t})\n\\]\nfor all $t \\geq 0.$\n\n\\subsection{Parking strategies and the space-based model}\n\nIn the model we have defined, all cars try to park as soon as they reach a free parking space. This can be thought of as a \\em parking strategy\\em. Let $G$ denote this ``greedy\" parking strategy: a car parks as soon as it can. It will be useful to consider different (possibly random) parking strategies as a way of controlling where cars park. In the definition below we introduce parking strategies more formally; $S_t(v,w)=1$ should be thought of as the event that the car starting from $v$ parks in $w$ at time $t.$\n\n\\begin{defn}\\label{parkdefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{parking strategy} $S = (S_t(v,w))_{t\\ge 1, v,w \\in V}$ for the car-based model on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $S_t(v,w)$ is $\\mathcal{F}_t$-measurable for each $v,w \\in V$ and $t \\ge 1.$\n\\item $\\sum_{t\\ge 1, w \\in \\mathbb{Z}} S_t(v,w) \\le 1$ (a car parks at most once).\n\\item $\\sum_{t \\ge 1, v \\in \\mathbb{Z}} S_t(v,w) \\le 1$ (a parking space can hold only one car).\n\\item $S_t(v,w) = 0$ whenever $B^v = 0$ (a parking space cannot be filled by a non-existent car).\n\\item $S_t(v,w) = 0$ whenever $B^w = 1$ (a car cannot park where there is no parking space).\n\\item $S_t(v,w) = 0$ whenever $X^v_t \\neq w$ (a car cannot park in a space which is not its current position).\n\\end{itemize}\nA car starting at $v$ \\em parks in space $w$ at time $t$ \\em if and only if $S_t(v,w) = 1.$\n\\end{defn}\nNote that $S_t(v,w)$ being $\\mathcal{F}_t$-measureable means that our parking strategy is previsible, and that the parking time of a car is a stopping time.\n\nFor a parking strategy $S$ and an event $E$ we let $\\mathbb{P}^S\\left[E\\right]$ denote the probability of $E$ when all cars follow strategy $S$ (note that $\\mathbb{P} = \\mathbb{P}^G$). We will also allow random parking strategies, which require suitable adjustments to the $\\sigma$-algebra and the filtration (for example, we may independently flip a coin at the start and choose different parking strategies depending on whether the coin is heads or tails).\n\nEquipped with these new definitions, we are nearly ready to prove Theorem \\ref{thm:supermarket}. The final element we shall need is a stochastically equivalent parking process, where the moves of cars are attached to spaces rather than the cars; we shall refer to this model as the \\emph{space-based parking model}.\n\n\\begin{defn}\\label{defn:space parking}\nLet $(L,R,\\mu)$ be a car park triple. Independently for each vertex $v \\in V$, let: \n\\begin{itemize}\n\\item $(E^v_n)_{n \\in \\mathbb{N}}$ be a sequence of independent $\\mu$-random variables,\n\n\\item $(\\tilde{U}^v_s)_{s \\in \\mathbb{N}}$ be a sequence of independent $\\mathrm{Unif}([0,1])$ random variables. \n\n\\item $\\tilde{B}^v$ be a $\\mbox{Bernoulli}(p)$ random variable. We initially place a car at $v$ when $\\tilde{B}^v = 1$ and otherwise a parking space with the capacity for one car.\n\\end{itemize}\n\nWhen a single car arrives (but does not park) at position $v$, it leaves in the next time step according to the first unused $E^v_n$. If the set of cars $\\{w_1,\\ldots,w_r\\}$ arrives at $v$ at time $s$ and do not park, they collect the next $r$ unused directions $E^v_n, E^v_{n+1}, \\ldots, E^v_{n+r-1}$, in the order determined by their increasing values of $U^{w_{\\ell}}_s$.\n\\end{defn}\n\nFor the space-based parking model on the car park triple $(L,R,\\mu)$, the probability space $(\\tilde{\\Omega},\\tilde{\\mathcal{F}},(\\tilde{\\mathcal{F}}_t)_{t\\ge0},\\tilde{\\mathbb{P}})$ is slightly less obvious than in the car-based parking model. This is because the number of directions collected from $E^v$ by cars that visit $v$ by time $t$ but do not park there depends on the behaviour of cars starting at distance at most $t$ from $v$ in the first $t$ steps of the process. Hence, we can define the filtration $\\left(\\tilde{\\mathcal{F}}_t\\right)_{t \\ge 0}$ to be\n\\[\n\\tilde{\\mathcal{F}}_t = \\sigma((\\tilde{B}^v)_{v \\in V}, (E^v_n)_{v \\in V, n < \\infty}, (\\tilde{U}^v_s)_{v \\in V,1 \\leq s \\leq t})\n\\]\nfor all $t \\geq 0$.\n\n\\begin{defn}\\label{spaceparkdefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{parking strategy} $\\tilde{S} = (\\tilde{S}_t(v,w))_{t\\ge 1, v,w \\in V}$ for the space-based model on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $\\tilde{S}_t(v,w)$ is $\\tilde{\\mathcal{F}}_t$-measurable for each $v,w \\in V$ and $t \\ge 1$.\n\\item $\\sum_{t\\ge 1, w \\in \\mathbb{Z}} \\tilde{S}_t(v,w) \\le 1$ (a car parks at most once).\n\\item $\\sum_{t \\ge 1, v \\in \\mathbb{Z}} \\tilde{S}_t(v,w) \\le 1$ (a parking space can hold only one car).\n\\item $\\tilde{S}_t(v,w) = 0$ whenever $\\tilde{B}^w = 1$ (a car cannot park where there is no parking space).\n\\item $\\tilde{S}_t(v,w) = 0$ whenever $\\tilde{B}^v = 0$ (a parking space cannot be filled by a non-existent car).\n\\item For all $v \\in L$ such that:\n\\begin{itemize}\n\\item $\\tilde{B}^v = 1$, and\n\\item for all $u \\in L$ and $s \\leq t-1$ we have $\\tilde{S}_s(v,u) = 0$,\n\\end{itemize}\nlet $E^{v_1}_{n_1}, E^{v_2}_{n_2}, \\ldots, E^{v_t}_{n_t}$ be the directions selected by $v$ in the first $t$ steps of its walk (note that we have $v_1 = v$). Then $\\tilde{S}_t(v,w) = 0$ if the walk obtained by starting at $v$ and following these directions does not end at $w$\n(a car cannot park in a space which is not its current position).\n\\end{itemize}\nA car starting at $v$ \\em parks in space $w$ at time $t$ \\em if and only if $\\tilde{S}_t(v,w) = 1.$\n\\end{defn}\n\nWe let $\\tilde{G}$ denote the greedy parking strategy in the space-based model.In the following proposition we show that parking strategies in the car-based parking process are stochastically equivalent to corresponding parking strategies in the space-based parking proces.\n\n\\begin{prop}\n\\label{prop:sameOld}\nLet $(L,R,\\mu)$ be a car park triple. Let $S$ and $\\tilde{S}$ be parking strategies for the car-based model and the space-based model on $(L,R,\\mu)$ respectively, and assume that for all $t \\geq 1$ and $v, w \\in L$ we have $S_t(v,w) = \\tilde{S}_t(v,w)$ whenever the following conditions hold:\n\\begin{enumerate}\n\\item $B^v = \\tilde{B}^v$ for all $v \\in L$ (the same cars appear in both models),\n\\item for all $1 \\leq s < t$ and $v, w \\in L$ we have $S_s(v,w) = \\tilde{S}_s(v,w)$ (at every time $1 \\leq s < t$, the same cars park in the same parking places in both models), and \n\\item for all $1 \\leq s \\leq t$, every car that does not park before time $s$, occupies the same position at time $s$ in both models\n\\end{enumerate}\n(i.e. the strategies $S$ and $\\tilde{S}$ behave identically whenever the cars behave identically up to time $t$ in the two processes). Then for any two sets $X \\subset L \\times L \\times \\mathbb{N}, Y \\subset L$, and the event\n\\[\n A_{X, Y} = [ \\mbox{ for all } (v_i, w_i, t_i) \\in X, \\mbox{car } v_i \\mbox{ is in } w_i \\mbox{ at time } t_i; \\mbox{ for all } w_j \\in Y, w_j \\mbox{ is a parking space } ]\n\\]\nwe have $\\mathbb{P}^S[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{S}}[A_{X,Y}]$.\n\\end{prop}\n\\begin{proof}\nWe have $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] \\leq (1-p)^{|Y|}$, so if $|Y| = \\infty$ then $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] = 0$ and the proposition holds.\n\nIf $|X| = \\infty$ then $A_{X,Y}$ must either describe the moves of infinitely many cars, or there must be a car $v$ such that $A_{X,Y}$ gives the position of $v$ at infinitely many times, or there are some $w_1 \\neq w_2$ and some $v \\in L, t \\in \\mathbb{N}$, such that $(v, w_1, t), (v, w_2, t) \\in X$. In all of these cases we have $\\mathbb{P}^S[A_{X,Y}], \\mathbb{P}^{\\tilde{S}}[A_{X,Y}] = 0$.\n\nHence we can assume that $|X|, |Y| < \\infty$. Then, let\n\\[\nU = \\{v : (v,w,t) \\in X \\} \\cup \\{w : (v,w,t) \\in X \\} \\cup Y,\n\\]\nand let $T = \\max \\{t : (v,w,t) \\in X\\}$. Then, in the car-based model, we can express $A_{X,Y}$ as a finite union of finite events concerning the variables $B^v, X^v_t, U^v_t$, for $t \\leq T$ and $v$ at distance at most $T$ from some element in $U$, describing the car\/parking space status and the step-by-step moves of cars in the $T$-neighbourhood of the elements if $U$. Analogously, in the space-based model, we can express $A_{X,Y}$ as a finite union of finite events concerning the variables $\\tilde{B}^v, E^v_n, \\tilde{U}^v_t$, for $t \\leq T$, $n \\leq T^2$, and $v$ at distance at most $T$ from some element in $U$. The proposition now follows from the properties of $S$ and $\\tilde{S}$, from the identical distributions and independence of $(B^v)_{v \\in V}$ and $(\\tilde{B}^v)_{v \\in V}$, of the $(U^v_t)_{v \\in V,t \\geq 0}$ and $(\\tilde{U}^v_t)_{v \\in V,t \\geq 0}$, as well as of $(X^v)_{v \\in \\mathbb{Z}}$ and $((E^v_n)_{n \\in \\mathbb{N}})_{v \\in \\mathbb{Z}}$ (observe that each of $E^v_n$ is used at most once in the process).\n\\end{proof}\n\nProposition \\ref{prop:sameOld} will allow us to deduce Theorem \\ref{thm:supermarket}  from the following theorem.\n\n\\begin{thm}\\label{thm:supermarketSpaces}\nLet $\\tilde{S}$ be a parking strategy for the space-based parking process on the car park triple $(L,R,\\mu)$. For a vertex $v$, we write $V^v_{\\tilde{S}}(t)$ for the value of $V^v(t)$ when strategy $\\tilde{S}$ is followed, and $V^v_{\\tilde{G}}(t)$ for the value of $V^v(t)$ when the greedy strategy is followed. Then for all $t \\ge 0,$ we have $V^v_{\\tilde{S}}(s) \\geq V^v_{\\tilde{G}}(s)$.\n\\end{thm}\n\n\\begin{proof}\nConsider the space-based parking process on a parking triple $(L,R,\\mu)$. Let $T^{v,r^{-1}}(t-1)$ be the number of cars that arrived at $vr^{-1}$ in the first $t-1$ time steps and then picked up $E^{vr^{-1}}_n = r$. Observe that $V^v(t)$ is equal to the sum over $r\\in R$ of $T^{v,r^{-1}}(t-1)$, plus $1$ if a car started at $v$ initially. By induction on $t$ we prove the following claim: for all $t \\geq 0$ we simultaneously have $T^{v,r^{-1}}_{\\tilde{S}}(t-1) \\geq T^{v,r^{-1}}_{\\tilde{G}}(t-1)$ and $V^v_{\\tilde{S}}(t) \\ge V^v_{\\tilde{G}}(t),$ for all $r \\in R$ (where again $T_{\\tilde{S}}$ and $T_{\\tilde{G}}$ denote the quantities when all cars follow strategy ${\\tilde{S}}$ or ${\\tilde{G}}$ respectively).\n\nIf a car parks at $v$ in the first $t$ time steps under ${\\tilde{S}}$ then $v$ must have initially been a parking space; then, if at least one car drove to $v$ under ${\\tilde{G}}$, it follows that some car parked in $v$ under ${\\tilde{G}}$ as well. Hence if the number of cars arriving at any vertex in the first $t$ time steps is at least as large under ${\\tilde{S}}$ as under ${\\tilde{G}}$, the same applies to the number of cars leaving $v$ in the first $t+1$ time steps. Moreover, for each $r \\in R,$ since the directions $E^{vr^{-1}}_n$ are selected one-by-one in a fixed order, $V^{vr^{-1}}_{\\tilde{S}} (t) \\ge V^{vr^{-1}}_{\\tilde{G}} (t)$ implies $T^{v,r^{-1}}_{\\tilde{S}} (t) \\geq T^{v,r^{-1}}_{\\tilde{G}} (t).$\n\nThe base case $t = 0$ of the induction is trivial. Hence suppose that our claim is true for $t = s -1 \\ge 0.$  By induction, for each $r \\in R,$ we have $V^{vr^{-1}}_{\\tilde{S}}(s-1) \\ge V^{vr^{-1}}_{\\tilde{G}}(s-1)$; hence we have $T^{v,r^{-1}}_{\\tilde{S}}(s-1) \\geq T^{v,r^{-1}}_{\\tilde{G}}(s-1).$ We then obtain\n\\begin{align*}\nV^v_{\\tilde{S}}(s) & = \\sum_{r\\in R} T^{v,r^{-1}}_{\\tilde{S}}(s-1) + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}} \\\\\n & \\geq \\sum_{r\\in R} T^{v,r^{-1}}_{\\tilde{G}}(s-1) + \\mathds{1}_{\\{v \\mbox{ \\small is a car}\\}} \\\\\n & = V^v_{\\tilde{G}}(s).\n\\end{align*}\nThis completes the proof of Theorem \\ref{thm:supermarketSpaces}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem  \\ref{thm:supermarket}] \nLet $S$ be a parking strategy for the car-based model on the car park triple $(L,R,\\mu)$, let $v \\in L$, and let $t, k \\ge 0$. Observe that for parking strategies in the space-based model, the filtration $\\tilde{\\mathcal{F}}_t$ carries all the information about the moves of all cars up to time $t$ (and, in fact, through $(E^v_n)_{v \\in V, n < \\infty}$ also about the movements to come). Therefore we can design a parking strategy $\\tilde{S}$ for the space-based model, such that the assumptions of Proposition \\ref{prop:sameOld} are satisfied for $S$ and $\\tilde{S}$.\n\nNext, we can express the event $[V^v_S(t) \\le k]$ as a finite union of events $A_{X,Y}$, defined as in Proposition \\ref{prop:sameOld}, describing the car\/parking space status and movements of cars starting at distance at most $t$ from $v$, such that at most $k$ cars arrive at $v$ by time $t$ under $S$. By Proposition \\ref{prop:sameOld}, we have $\\mathbb{P}^S[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{S}}[A_{X,Y}]$. By Theorem \\ref{thm:supermarketSpaces} we have $V^v_{\\tilde{S}}(s) \\geq V^v_{\\tilde{G}}(s)$ deterministically, hence if $A_{X,Y} \\subseteq [V^v_{\\tilde{S}}(t) \\le k]$, then also $A_{X,Y} \\subseteq [V^v_{\\tilde{G}}(t) \\le k]$. Thus we have $\\mathbb{P}[V^v_{\\tilde{G}}(t) \\le k] \\geq \\tilde{\\mathbb{P}}[V^v_{\\tilde{S}}(t) \\le k]$, and since by applying Proposition \\ref{prop:sameOld} again we find that $\\mathbb{P}^G[A_{X,Y}] = \\tilde{\\mathbb{P}}^{\\tilde{G}}[A_{X,Y}]$, we finally obtain $\\mathbb{P}[V^v_G(t) \\le k] \\geq \\mathbb{P}[V^v_S(t) \\le k]$ as claimed.\n\\end{proof}\n\nIn the rest of this paper, we shall consider the car-based parking model only. We remark that for parking times we may not make a conclusion similar to Theorem \\ref{thm:supermarket}. For example, consider the parking strategy where all but one car is instructed to never park. The chosen car will have a much easier job of finding a parking space. To combat this, we need some symmetry that will allow us compare visits to a space and parking times of cars, and therefore make use of Theorem \\ref{thm:supermarket}\n\nWe say that a parking strategy $S$ on the car park triple $(L,R,\\mu)$ is \\emph{weakly translation invariant} if for all $v,w \\in V$, $r\\in R$  and $t \\ge 0$, \n\\[\n \\mathbb{P}^S\\left[S_t(v,w)=1\\right] = \\mathbb{P}^S\\left[S_t(vr,wr)=1\\right].\n\\]\nAn equivalent property is that for all $v,w \\in V$, $r\\in R$ and $t \\geq 1$,\n\\[\n\\mathbb{P}^S\\left[\\mbox{car $v$ arrives at spot $w$ at time $t$}\\right] = \\mathbb{P}^S\\left[\\mbox{car $vr$ arrives at spot $wr$ at time $t$}\\right].\n\\] \n\n\\begin{rem}\nThis is a rather weak form of translation invariance -- it does not control joint events in any sense. Since in this paper we are predominantly working with expectations, we do not need to worry about this. A more natural form of translation invariance is the following form: a parking strategy $S$ on the car park triple $(L,R,\\mu)$ is \\emph{strongly translation invariant} if for any $r\\in R,$ the probability measure $\\mathbb{P}$ is invariant with respect to a translation by $r$. (The same is true for car removal strategies which we introduce later.) We note that the parking strategy (respectively, car removal strategy) we use in Section \\ref{sec:p=1\/2upper} (respectively, Section \\ref{sec:p=1\/2lower}) are in fact strongly translation invariant.\n\\end{rem}\n\nWeak translation invariance allows us to equate car journey lengths with total number of visits to a position in $V.$\n\n\\begin{lem}\\label{lem:dual}\nLet $S$ be a weakly translation invariant strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0$ and $v \\in V,$\n\\[\n \\mathbb{E}^S[\\tau \\wedge t] = \\mathbb{E}^S[V^v(t)].\n\\]\n\\end{lem}\n\nWe remark that this result is a special case of the well known and more general Mass-Transport principle \\cite[Theorem 8.7]{L+P} and a similar result was noted at \\cite[Lemma 4.1]{DGJLS}. Since the proof is very short in our setting, we include it for self-containment.\n\n\\begin{proof}\nLet $t \\ge 0$ and fix an arbitrary $v \\in V.$ Write $B_t(v)$ for the vertices of $L$ connected to $v$ by a path of length at most $t$. By translation invariance\n\\begin{align*}\n \\mathbb{E}^S[\\tau \\wedge t] = \\mathbb{E}^S[\\tau^v \\wedge t] & = \\sum_{s \\in [t]}\\sum_{w \\in B_{t}(0)} \\mathbb{P}^S\\left[\\mbox{car $v$ arrives at spot $vw$ at time $s$}\\right] \\\\\n & = \\sum_{s \\in [t]}\\sum_{w \\in B_t(0)} \\mathbb{P}^S\\left[\\mbox{car $vw^{-1}$ arrives at spot $v$ at time $s$}\\right] \\\\\n & = \\mathbb{E}^S[V^v(t)].\n\\end{align*}\n\\end{proof}\n\nThe following easy corollary of Theorem \\ref{thm:supermarket} and Lemma \\ref{lem:dual} is crucial for our arguments, and considers the expected journey of a car up to time $t$ under different parking strategies. It will allow us to derive upper bounds on $\\mathbb{E}^G[\\tau \\wedge t]$ by considering a different parking strategy which is easier to control.\n\n\\begin{cor}\n\\label{lem:supermarket}\nLet $S$ be a weakly translation invariant parking strategy on the car park triple $(L,R,\\mu)$. Then for all $t \\ge 0,$\n\\[\n \\mathbb{E}^S[\\tau \\wedge t] \\ge \\mathbb{E}^G[\\tau \\wedge t].\n\\]\n\\end{cor}\n\n\\subsection{Car removal strategies}\n\nAnother way to modify the car parking problem is through \\emph{car removal strategies}. Under certain circumstances it will be helpful to pretend that a car has been removed from the process. A car is removed during a step, and it is parked off $V$. So if car $v$ is at position $w$ at time $t,$ and is removed during step $t+1$, we remove the car from the process without it taking up a parking space and set $\\tau^v = t+1.$ We remark that we will always assume a greedy parking strategy when we have a non-trivial car removal strategy.\n\n\\begin{defn}\\label{teledefn}\nLet $(L,R,\\mu)$ be a car park triple. A \\emph{car removal strategy} $Q = (Q_t(v))_{t\\ge 1, v\\in V}$ on $(L,R,\\mu)$ is a sequence of random variables taking values in $\\{0,1\\}$ with the following properties:\n\\begin{itemize}\n\\item $Q_t(v)$ is $\\mathcal{F}_t$-measurable for each $v \\in V$ and $t \\ge 1.$\n\\item $Q_t(v) = 0$ whenever $B^v = 0$ (a non-existent car cannot be removed).\n\\item $\\sum_{t\\ge 1} Q_t(v) \\le 1$ (a car can only be removed once).\n\\end{itemize}\nA car starting at $v$ is removed in the $t$-th time step if and only if $Q_t(v) = 1.$\n\\end{defn}\nAs we did for parking strategies, we define $\\mathbb{P}^Q$ for a car removal strategy $Q$. Whenever we explicitly consider a process involving car removal strategies, we assume that all vehicles follow the greedy parking strategy.\n\nWe are now ready to prove Theorem \\ref{thm:pinkfloyd}.  In the one-dimensional setting this will allow us to derive lower bounds on $\\mathbb{E}[\\tau \\wedge t]$ by considering an interval and removing cars that enter or leave the interval.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:pinkfloyd}.]\nFor each $w \\in V$ and $t \\ge 0,$ let $W^w_Q(t)$ be the set of unparked cars at position $w$ at time $t$ under $Q$, and let $W^w_G(t)$ denote the same quantity under $G$ (recall that under $G$, which is the greedy parking strategy, there is no car removal). We start by showing that at every position $w \\in V$ and for every time $t \\ge 0$ we have $W^w_Q(t) \\subseteq W^w_G(t)$. We prove this by induction on $t \\ge 0$. The base case $t=0$ is trivial, hence suppose that the claim is true up to and including time $t-1$.\n\nFix a position $w$ and observe first that if a parking space $w$ is filled at time $t$ under $Q$, then a car $v$ from $W^{wr^{-1}}_Q(t-1)$ must arrive at $w$ at time $t$ for some $r \\in R$. By the inductive hypothesis, $v$ must be in the appropriate set in $W^{wr^{-1}}_G(t-1)$, and so it must arrive at $w$ at time $t$ under $G$ (note that we are in the original parking process, where cars have random walks attached to them, rather than the space-based parking process considered in the proof of Theorem \\ref{thm:supermarketSpaces}). Therefore under $G$ either spot $w$ must already be filled before $t$, or a car must park in spot $w$ at time $t$. Therefore any parking space filled under $Q$ at time $t$ must be filled under $G$ at time not later than $t$.\n\nNow, by the inductive hypothesis, any car arriving at position $w$ under $Q$ at time $t$ must arrive at position $w$ under $G$ at time $t$. If $w$ is not a free parking space under $G$ at time $t-1$, then $W^w_Q(t) \\subseteq W^w_G(t)$ and the claim holds. Thus suppose that $w$ is a free parking space at time $t-1$ under $G$. Then by the argument above, $w$ must be a free parking space at time $t-1$ under $Q$. Further, if under $G$ a car not from $W^w_Q(t)$ parks at $w$ at time $t$, then again $W^w_Q(t) \\subseteq W^w_G(t)$ and again we are done. So suppose that under $G$ a car $v \\in W^w_Q(t)$ parks at position $w$ at time $t$. By the tie-breaking procedure, $v$ must have the smallest $U^x_t$ value over the cars $x$ that arrive at $w$ under $G$, and so must have the smallest $U^x_t$ value over cars $x$ that arrive at $w$ under $Q$. Therefore under $Q$ the car $v$ must also park at $w$ at time $t$, and so once again we have $W^w_Q(t) \\subseteq W^w_G(t)$.\n\nNow, consider that the set of unparked cars at time $t$ is the union $\\bigcup_{w \\in V}W^w(t)$, hence if a car $v$ is still unparked under $Q$ at time $t$, then there is some $w \\in L$ such that $v \\in W^w_Q(t)$. But we know that $W^w_Q(t) \\subseteq W^w_G(t)$, therefore we have $v \\in W^w_G(t)$, implying that $\\tau^v_Q \\le \\tau^v_G$ as desired.\n\nAlso, the number of visits to $w$ at time $t$ is\n\\[\n|W^w(t)|+ \\mathds{1}_{\\{\\mbox{\\small a car parks at } w \\mbox{ \\small at time } t\\}}.\n\\]\nSince $W^w_Q(t) \\subseteq W^w_G(t)$, and additionally we know that\n\\[\n\\sum_{s=1}^t \\mathds{1}_{\\{\\mbox{\\small a car parks under } Q \\mbox{ at } w \\mbox{ \\small at time } s\\}} \\leq \\sum_{s=1}^t \\mathds{1}_{\\{\\mbox{\\small a car parks under } G \\mbox{ at } w \\mbox{ \\small at time } s\\}},\n\\]\nthe inequality $V^v_Q(t) \\le V^v_G(t)$ follows.\n\\end{proof}\n\n\\section{Probabilistic bounds}\n\\label{sec:probabilities}\nIn this section, we state some probabilistic bounds that are needed for the proofs in Sections \\ref{sec:p=1\/2upper}, \\ref{sec:p=1\/2lower}, and \\ref{sec:p<1\/2}. \n\nWe make use of the following variant of the Chernoff bound (see \\cite[Chapter~4]{Chernoffcite}).\n\n\\begin{lem}\\label{feelthechern}\nLet $p \\in (0,1),$ $N \\in \\mathbb{N},$ and $\\varepsilon > 0.$ Then $$\\mathbb{P}\\left[\\Bin(N,p) \\ge N(p+\\varepsilon)\\right] \\le e^{-2\\varepsilon^2N}.$$\n\\end{lem}\n\nWe need some facts about hitting times of the simple symmetric random walk.\n\\begin{lem}\\label{lbeforer}\nLet $a, b > 0$ be positive integers. Let $\\{X_n\\}_{n \\geq 0}$ be a simple symmetric random walk on $\\mathbb{Z}$ with $X_0 = 0.$ For $i \\in \\mathbb{Z},$ let $H_i = \\min\\{s : X_s = i\\}.$ Then\n\t\\begin{itemize}\n\t\t\\item[(i)] $\\mathbb{P}[H_b < H_{-a}] = \\frac{a}{a+b}.$ \n\t\t\\item[(ii)] $\\mathbb{E}[H_b | H_b < H_{-a}] = \\frac{b(b+2a)}{3}.$\n\t\t\\item[(iii)] $\\mathbb{E}[H_{-a} \\wedge H_{a}] = a^2.$\n\t\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nAll of this is standard. Part (i) is Gambler's ruin (see \\cite[XIV.2]{Feller-Probability}). Part (iii) follows from (ii) by symmetry and a simple calculation.\n\nFor part (ii), we first prove the statement in a slightly different setup. Let $c,d$ be positive integers with $0 < c < d$ and assume that $X_0 = c.$ We show that\n\\[\n\\mathbb{E}[H_d | H_d < H_0] = \\frac{(d-c)(d+c)}{3}.\n\\]\nPart (ii) of the lemma then follows immediately by taking $d = a+b$ and $c=a$. Let $Z_n = X_n^3 - 3nX_n$ and let $S =  X_{n+1}-X_n \\in \\{-1,1\\}$. Then, since $S^2 = 1$ and $S^3 = S$, we have\n\\begin{align*}\nZ_{n+1} & = \\left(X_n+S\\right)^3 - 3(n+1)\\left(X_n+S\\right) \\\\\n & = Z_n + 3X_n^2S+3X_n S^2+S^3 - 3nS - 3X_n -3 S \\\\\n & = Z_n + S\\left(3X_n^2-2-3n\\right).\n\\end{align*}\n\nSince $X_{n+1}-X_n$ takes values in $\\{-1,+1\\}$ with mean $0$ independently of $\\mathcal{F}_n,$ and since $X_n$ is $\\mathcal{F}_n$-measurable, we have\n\\begin{align*}\n\t\t\\mathbb{E}[Z_{n+1} | \\mathcal{F}_n] & = \\mathbb{E}[Z_n + \\left(X_{n+1}-X_n\\right)\\left(3X_n^2-2-3n\\right) | \\mathcal{F}_n] \\\\\n\t\t& = Z_n + \\left(3X_n^2-2-3n\\right)\\mathbb{E}[X_{n+1}-X_n] = Z_n,\n\\end{align*}\nand so $Z$ is a martingale.\n\nFor $n \\in \\mathbb{N},$ Doob's Optional Stopping Theorem gives $\\mathbb{E}[Z_{n\\wedge H_0 \\wedge H_d}] = \\mathbb{E}[Z_0] = c^3.$ At the same time, $|Z_{n\\wedge H_0 \\wedge H_d}|$ is bounded by $3d^3 + 3(H_0 \\wedge H_d)d$ for all $n.$\nAdditionally, $H_0 \\wedge H_d$ is integrable and so by the Dominated Convergence Theorem we have\n\\[\n  \\mathbb{E}[Z_{H_0 \\wedge H_d}] = \\lim_{n \\to \\infty} \\mathbb{E}[Z_{n\\wedge H_0 \\wedge H_d}] = c^3.\n\\]\nBut $Z_{H_0 \\wedge H_d} = \\mathds{1}_{H_d < H_0}(d^3 -3dH_d).$ Therefore\n\t\\begin{align*}\n\t\tc^3 = \\mathbb{E}[Z_{H_0 \\wedge H_d}] & = \\mathbb{E}[\\mathds{1}_{H_d < H_0}(d^3 -3dH_d)]  \\\\\n\t\t& = \\mathbb{P}\\left[H_d < H_0\\right](d^3 - 3d\\mathbb{E}[H_d | H_d < H_0]).\n\t\\end{align*}\nBy (i), $\\mathbb{P}\\left[H_d < H_0\\right] = c\/d,$ and so $\\mathbb{E}[H_d | H_d < H_0] = \\frac{d^3 - c^2d}{3d} = \\frac{d^2-c^2}{3}.$\n\\end{proof}\n\n\nLet $M_n$ denote the maximum value in the first $n$ time steps of the simple symmetric random walk starting at $0,$ and let $m_n$ denote its corresponding minimum value. Define $p_{n,r} = \\binom{n}{\\frac{n+r}{2}} 2^{-n}.$ It can be shown (see, e.g., Feller \\cite[Theorem III.7.1]{Feller-Probability}) that for $r \\geq 0$ we have\n\\[\n \\mathbb{P}\\left[M_n = r\\right] = \\mathbb{P}\\left[m_n = -r\\right] = \\begin{cases}\np_{n,r} \\quad & \\mbox{if $n - r$ is even}, \\\\\np_{n,r+1} \\quad & \\mbox {otherwise.}\n\\end{cases}\n\\]\n\nLet $Y \\sim \\Bin(n,1\/2)$, where we will assume that $n$ is even, so that for $k \\le n\/2,$ $\\mathbb{P}\\left[Y = \\frac{n+2k}{2}\\right] = p_{n,2k}.$ We now conclude this section with some tail bounds for the maximum of the random walk.\n\n\\begin{lem}\\label{minssrw}\n\\begin{itemize}\n\\item[(i)] $\\mathbb{P}\\left[M_n \\ge 2 \\alpha  \\sqrt{n \\log n}\\right] \\le 2 n^{-2 \\alpha^2}.$\n\\item[(ii)] $\\mathbb{P}\\left[M_n \\ge c\\sqrt{n}\\right]$ and $\\mathbb{P}\\left[M_n \\le c\\sqrt{n}\\right]$ are bounded away from zero for each $c>0.$\n\\end{itemize}\n\\end{lem}\n\nWe remark that the analogous results hold for $m_n$ by symmetry.\n\n\\begin{proof}\n\t\\begin{align}\n\t\t\\mathbb{P}\\left[M_n \\ge 2k\\right] & = \\mathbb{P}\\left[M_n = 2k\\right] + \\sum_{\\ell = k+1}^{n\/2} \\left ( \\mathbb{P}\\left[M_n = 2\\ell-1\\right] + \\mathbb{P}\\left[M_n=2\\ell \\right] \\right ) \\nonumber \\\\\n\t\t& = p_{n,2k} + \\sum_{\\ell = k+1}^{n\/2} \\left ( p_{n,2\\ell-1} + p_{n,2\\ell} \\right ) \\nonumber \\\\\n\t\t& = p_{n,2k} + 2 \\sum_{\\ell = k+1}^{n\/2} p_{n,2\\ell} \\nonumber \\\\\n\t\t& = \\mathbb{P}\\left[Y = \\frac{n+2k}{2}\\right] + 2 \\sum_{\\ell=k+1}^{n\/2} \\mathbb{P}\\left[Y = \\frac{n+2\\ell}{2}\\right] \\label{double-ish} \\\\\n\t\t& \\leq 2\\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right]. \\nonumber\n\t\\end{align}\n\nThe same holds for odd $n,$ and so we see that $\\mathbb{P}\\left[M_n \\ge 2k\\right] \\le 2\\mathbb{P}\\left[Y \\ge n(1\/2 + k\/n)\\right].$ Setting $k = \\alpha \\sqrt{n\\log n}$ and applying Lemma \\ref{feelthechern} gives\n\t\\[\n\t\t\\mathbb{P}\\left[M_n \\ge 2 \\sqrt{n \\log n}\\right] \\le 2n^{-2 \\alpha^2}.\n\t\\]\n\nSetting $k = c\\sqrt{n}$ we get\n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[M_n \\le 2k\\right] & \\ge 1-\\mathbb{P}\\left[M_n \\ge 2k\\right]  \\\\\n\t\t& \\ge 1 - 2 \\mathbb{P}\\left[Y \\ge n(1\/2 + k\/n)\\right]  \\\\\n\t\t& = 1 - 2\\mathbb{P}\\left[\\frac{Y-n\/2}{\\sqrt{n\/4}} \\ge 2c\\right]  \\\\\n\t\t& \\rightarrow 1 - 2(1-\\Phi(2c)),   \\\\\n\t\t& = 2\\Phi(2c)-1, \n\t\\end{align*}\nas $n \\rightarrow \\infty$ by the Central Limit Theorem. Since $c > 0$ we have $\\Phi(c) > 1\/2,$ and so $\\mathbb{P}[M_n \\le c\\sqrt{n}]$ is bounded away from zero for each $c>0.$\n\nFrom \\eqref{double-ish}, we may read off $\\mathbb{P}\\left[M_n \\ge 2k\\right] \\ge 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k+2}{2}\\right] = 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right]-O(n^{-1\/2}),$ and so again for $k = c\\sqrt{n},$ \n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[M_n \\ge 2k\\right] & \\geq 2 \\mathbb{P}\\left[Y \\ge \\frac{n+2k}{2}\\right] -o(1) \\\\\n\t\t& = 2 \\mathbb{P}\\left[\\frac{Y-n\/2}{\\sqrt{n\/4}} \\ge 2c\\right] -o(1) \\\\\n\t\t& \\rightarrow 2-2\\Phi(2c) > 0 \n\t\\end{align*}\nas $n \\rightarrow \\infty$ by the Central Limit Theorem. Therefore $\\mathbb{P}\\left[M_n \\ge c\\sqrt{n}\\right]$ is bounded away from zero for each $c>0.$\n\\end{proof}\n\\section{Upper bound on $\\mathbb{E} [\\tau \\wedge t]$}\n\\label{sec:p=1\/2upper}\n\nIn this section, we prove the upper bound in Theorem \\ref{thm:p=1\/2}. We fix a target time $t$ and consider a particular weakly translation invariant parking strategy with additional properties. The parking strategy assigns (at time $0$) a parking space to most of the cars and tells the other cars they can never park. Each car then drives until it reaches its assigned parking place (or just keeps driving if it has no assigned space). The work left to do is to show that many cars are assigned parking spaces that they will reach in a short expected amount of time. We split this section into two parts; the first one detailing the parking strategy and showing some of its properties, and the second one bringing everything together to prove the desired upper bound.\n\n\\subsection{The parking strategy}\nFix $t \\ge 1.$ We define the parking strategy $T = T_t$ as follows. We first divide $\\mathbb{Z}$ into intervals of length $\\lceil \\sqrt{t} \\rceil.$ On each interval $I,$ we run through the locations from right to left, attempting to assign to each car $i$ a parking space $P(i)$ somewhere in $I$ and to the left of $i.$ If there is no unassigned parking space available within distance $O(t^{1\/4})$ then car $i$ will not try to park, and we set $P(i) = \\star;$ and if $i$ is a parking space we set $P(i) = i.$ This defines a strategy that is periodic, but not weakly translation invariant (because the intervals have specified endpoints). So we begin by applying a random shift to our intervals to make the strategy weakly translation invariant.\n\nMore formally, let $\\zeta = \\lceil \\sqrt{t} \\rceil$ and $\\nu = \\lceil t^{1\/4} \\rceil.$ First let $Z$ be uniformly distributed on $[\\zeta]$ independently from the original model. Then, given $Z=z,$ for each interval $[z+k\\zeta,z+(k+1)\\zeta-1]$ we assign specific parking spaces to cars as follows:\n\n\\begin{algorithm}[H]\n\\SetKw{KwFn}{Initialization}\n    \\KwFn{Set $m=z+(k+1)\\zeta-1,$ $W=\\emptyset$}\\;\n    \\While{$m \\ge z+k\\zeta$}{\n    \\If{\\rm{There is initially a parking space at $m$}}{\n    \t    Set $P(m) = m$\\;\n    \t    \\If{$W \\neq \\emptyset$}{\n            \tLet $v$ be the largest element of $W.$\\:\n            \tRemove $v$ from $W$ and set $P(v) = m$\\;\n            }\n        }\n      \\If{\\rm{There is initially a car at $m$}}{\n            Add $m$ to $W$\\;\n        }\n     \\If{$|W| = \\nu$}{\n            Let $v$ be the largest element of $W.$\\:\n            Remove $v$ from $W$ and set $P(v) = \\star$\\;\n        }\n    Set $m := m-1$\\;\n    }\n\\SetKw{KwFm}{Finalization}\n  \\KwFm{For all $v \\in W,$ set $P(v) = \\star.$}\n\\label{discardalgorithmpark}\n\\end{algorithm}\n\\bigskip\n\nThe strategy $T$ is defined as follows: for each car $i,$\n\\begin{itemize}\n\\item if $P(i) = \\star,$ then $S_t(i,j)=0$ for all $t \\ge 1,$ $j \\in \\mathbb{Z}$ (car $i$ never parks).\n\\item if $P(i) \\neq \\star,$ then $S_t(i,P(i)) = 1$ for the first time $t$ when car $i$ visits $P(i),$ and $S_t(i,j) = 0$ otherwise.\n\\end{itemize}\n\nNote that the random variable $Z$ causes this parking strategy to be weakly translation invariant, and so it is sufficient to show that $\\mathbb{E}^T[\\tau \\wedge t] = O(t^{3\/4})$ to prove the upper bound in Theorem \\ref{thm:p=1\/2}.\n\nThe benefit of this parking strategy is that it is much easier to give bounds on the expected hitting time of a fixed vertex rather than an arbitrary empty parking space. However, there are a couple of potential problems: the parking strategy might assign cars to distant parking spaces; and the parking strategy might dictate that many cars never park ($P(v) = \\star$ for too many $v$). The next two lemmas resolve these problems.\n\n\\begin{lem}\\label{sorting}\nFor all $i$ we have $P(i) = \\star$ or $i-P(i) \\le 3\\nu.$\n\\end{lem}\n\n\\begin{lem}\\label{markov}\nFor all $i \\in \\mathbb{Z},$ $\\mathbb{P}\\left[P(i) = \\star\\right]  = O(t^{-1\/4}).$\n\\end{lem}\n\nLemma \\ref{sorting} follows from our choice to abandon the oldest car when the queue is too long.\n\\begin{proof}[Proof of Lemma \\ref{sorting}]\nSince we define $P(i) = i$ whenever $i$ is a parking space, the lemma is equivalent to saying that a vertex does not stay in $W$ for too long. Indeed suppose that $i$ joins $W.$ Since $W$ contains at most $\\nu$ elements at any time and elements of $W$ get assigned a parking space (or $\\star$) in the order of when they join $W,$ if $P(i) \\neq \\star,$ then $i$ will be assigned a parking space $P(i)$ with $i-P(i) \\le 3\\nu$ if there are at least $\\nu$ parking spaces in the next $3\\nu$ iterations of the while loop of the algorithm above.\n\nSuppose that there are fewer than $\\nu$ parking spaces in the next $3\\nu$ iterations of the while loop. Then the number of elements joining $W$ is at least $2\\nu.$ Since $W$ contains at most $\\nu$ elements at any time, at least $\\nu$ elements must leave $W$ during these $3\\nu$ iterations of the while loop. Therefore $i$ must leave $W$ during these $3\\nu$ iterations of the loop, either with $i-P(i) \\leq 3\\nu$ or with $P(i) = \\star.$\n\\end{proof}\n\nThe proof of Lemma \\ref{markov} is a little more involved. We use some elementary properties of irreducible, aperiodic Markov chains.\n\n\\begin{proof}[Proof of Lemma \\ref{markov}]\nWe may assume without loss of generality that $Z$ is $0$, and we consider the interval obtained by taking $k=0$. By symmetry and translation invariance, we see that for any $i \\in \\mathbb{Z}$\n\t\\begin{equation}\n\t\t\\mathbb{P}^T\\left[P(i) = \\star\\right] = \\zeta^{-1}\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|]. \\label{symsneak}\n\t\\end{equation}\n\nLet $C_n$ be the size of $W$ just before the last \\textbf{if} clause of the loop when $m = \\zeta-n,$ and set $C_0 = 0.$ In most situations we can only have $C_{n+1}-C_n$ equal to either 1 (if $\\zeta-n-1$ is a car) or $-1$ (if $\\zeta-n-1$ is a parking space). However, there are two exceptions to that rule. If $C_n = 0,$ i.e., if $W = \\emptyset$ after we observe $\\zeta-n,$ and if $\\zeta-n-1$ is a parking space, then $C_{n+1} = 0$ as well. Moreover, if $C_n = \\nu$ then in the last \\textbf{if} clause of the loop we deterministically remove one element from $W.$ Thus depending on the value of whether $\\zeta-n-1$ is a car o a parking space, we might have either $C_{n+1} = \\nu$ or $C_{n+1} = \\nu-2.$ Hence $C = (C_0, C_1, \\ldots)$ is a Markov chain with transition probabilities $(p_{k,l})_{k,l \\in \\{0,\\ldots,\\nu\\}}$ satisfying:\n\t\\begin{itemize}\n\t\t\\item $p_{0,0} = 1\/2$ (there is a parking space but no queue),\n\t\t\\item $p_{0,1} = 1\/2$ (a car joins an empty queue),\n\t\t\\item $p_{k,k-1} = 1\/2$ when $i \\in \\{1,\\ldots,\\nu-1\\}$ (a car in the queue is assigned a parking space),\n\t\t\\item $p_{k,k+1} = 1\/2$ when $i \\in \\{1,\\ldots,\\nu-1\\}$ (a new car joins the queue),\n\t\t\\item $p_{\\nu,\\nu-2} = 1\/2$ (we tell an old car to leave the queue, and assign another queueing car to a parking space),\n\t\t\\item $p_{\\nu,\\nu} = 1\/2$ (we tell an old car to leave the queue, and a new car joins the queue),\n\t\t\\item $p_{k,l} = 0$ otherwise.\n\t\\end{itemize}\n\t\nWe see that some vertex gets assigned $\\star$ each time $C$ hits $\\nu.$ Additionally, the $C_{\\zeta}$ vertices remaining in $W$ at the end of the execution of the algorithm also get assigned $P(v) = \\star.$ Therefore\n\t\\begin{eqnarray}\n\t\t\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right| = C_{\\zeta} + \\sum_{n=0,\\ldots,\\zeta-1} \\mathds{1}_{C_n = \\nu} \\label{compare}\n\t\\end{eqnarray}\nIn our algorithm, we initially impose that $W = \\emptyset.$ If, however, we started the algorithm with $W'$ containing some cars, then at every step in the algorithm, we would have $W \\subseteq W'.$ Let $C'_n$ be the size of $W'$ just before the last \\textbf{if} clause of the loop when $m = \\zeta - n.$ Then we see that $\\{C'_n\\}$ is a Markov chain with transition probabilities $(p_{k,l})_{k,l \\in \\{0,\\ldots,\\zeta\\}}$ which majorises $C.$ Thus, if $|W'|$ initially has distribution $\\mu,$ we see\n\t\\[\n\t\t\\mathbb{P}[C_n = \\nu] \\le \\mathbb{P}\\left[C'_n = \\nu\\right] = \\mathbb{P}_{C_0 \\sim \\mu}\\left[C_n = \\nu\\right].\n\t\\]\nIn particular, if we let $\\pi$ be a stationary distribution of $C,$ then for all $n$\n\t\\[\n\t\t\\mathbb{P}_{C_0=0}\\left[C_n = \\nu\\right] \\le \\mathbb{P}_{C_0 \\sim \\pi}\\left[C_n= \\nu\\right] = \\pi(\\nu).\n\t\\]\nHence if we take the expectation of \\eqref{compare} we obtain\n\t\\[\n\t\t\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|] \\le \\mathbb{E}^T[C_{\\zeta}] + \\zeta\\pi(\\nu).\n\t\\]\n\nSince $C$ is irreducible and aperiodic, and has a finite state space, it has a unique stationary distribution $\\pi.$ One can then verify that $\\pi(k) = \\frac{1}{\\nu}$ for $k=0,\\ldots \\nu-2,$ and $\\pi(k) = \\frac{1}{2\\nu}$ for $k = \\nu-1,\\nu.$ Since $C$ takes values in $0,\\ldots,\\nu,$ we may bound $\\mathbb{E}[X_{\\zeta}]$ by $\\nu$ to find\n\t\\[\n\t\t\\mathbb{E}^T[\\left|\\{j \\in [0,\\zeta-1] : P(j) = \\star\\}\\right|] \\le \\nu + \\frac{\\zeta}{2\\nu}.\n\t\\]\nTogether with \\eqref{symsneak} we obtain $\\mathbb{P}^T \\left[P(i) = \\star\\right] \\le \\frac{\\nu}{\\zeta} + \\frac{1}{2\\nu}= O(t^{-1\/4}).$\n\\end{proof}\n\n\\subsection{Proof of the upper bound}\nWe now have all the ingredients necessary to prove the upper bound in Theorem \\ref{thm:p=1\/2}. We will do this by bounding $\\mathbb{E}^T[\\tau \\wedge t]$ and then appealing to Corollary \\ref{lem:supermarket}.\n\\begin{proof}[Proof of the upper bound in Theorem \\ref{thm:p=1\/2}]\nLet $t \\ge 0.$ Without loss of generality we can consider $\\tau = \\tau^0.$ Then\n\t\\begin{align*}\n\t\t\\mathbb{E}^T[\\tau \\wedge t] &= \\mathbb{E}^T[\\tau^0 \\wedge t] \\\\\n\t\t&= \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) = \\star]\\mathbb{P}^T\\left[P(0) = \\star\\right] + \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star]\\mathbb{P}^T\\left[P(0) \\neq \\star\\right] \\\\\n\t\t&\\le t\\mathbb{P}^T\\left[P(0) = \\star\\right] + \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star].\n\t\\end{align*}\n\t\nLemma \\ref{markov} gives $\\mathbb{P}^T[P(v) = \\star] = O(t^{-1\/4})$ and so\n\t\\begin{equation}\n      \\label{uyp1}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le \\mathbb{E}^T[\\tau^0 \\wedge t | P(0) \\neq \\star] + O(t^{3\/4}).\n\t\\end{equation}\n\nLet $a = 3\\nu = 3 \\lceil t^{1\/4} \\rceil, b = \\zeta = \\lceil \\sqrt{t} \\rceil$. For an integer $m,$ let $H_m$ be the first hitting time of the random walk $X^0$ to $m.$ Lemma \\ref{sorting} tells us that if $P(0) \\neq \\star,$ then $P(0) \\ge -a.$ We therefore see $\\tau^0 \\wedge t = H_{P(0)} \\wedge t \\le H_{-a}.$ When $H_{-a} > H_b,$ we may trivially bound $\\tau^0 \\wedge t$ by $t.$ Putting this into \\eqref{uyp1} gives\n\t\\begin{align*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t]  \\le \\mathbb{E}^T[H_{-a} &| H_{-a} < H_b, P(0) \\neq \\star]\\mathbb{P}^T[H_{-a} < H_b | P(0) \\neq \\star] \\\\\n\t\t& + t\\mathbb{P}^T\\left[H_{-a} > H_b | P(0) \\neq \\star\\right] + O(t^{3\/4}).\n\t\\end{align*}\n\t\nClearly $X^0$ is independent from $P(0),$ which only depends on the initial configuration, and so\n\t\\begin{eqnarray*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le \\mathbb{E}[H_{-a} | H_{-a} < H_b] + t\\mathbb{P}\\left[H_{-a} > H_b\\right] + O(t^{3\/4}).\n\t\\end{eqnarray*}\n\t\nLemma \\ref{lbeforer} (i) and (ii) tells us that $\\mathbb{P}\\left[H_b<H_{-a}\\right] = O(t^{-1\/4})$ and $\\mathbb{E}[H_{-a} | H_{-a}<H_b] = O(t^{3\/4}).$ We therefore see that\n\t\\begin{eqnarray*}\n\t\t\\mathbb{E}^T[\\tau^0 \\wedge t] \\le O(t^{3\/4}) + tO(t^{-1\/4}) + O(t^{3\/4}) = O(t^{3\/4}).\n\t\\end{eqnarray*}\n\t\nFinally, we appeal to Corollary \\ref{lem:supermarket} to obtain\n\\[\n\\mathbb{E}[\\tau \\wedge t] = \\mathbb{E}^G[\\tau \\wedge t] \\leq \\mathbb{E}^T[\\tau \\wedge t] = O(t^{3\/4}).\n\\]\n\\end{proof}\n\n\n\\section{Lower bound on $\\mathbb{E}^T[\\tau \\wedge t]$}\n\\label{sec:p=1\/2lower}\n\nIn this section, we prove the lower bound in Theorem \\ref{thm:p=1\/2}. We do this by considering a parking process on an interval, and appealing to various properties of the simple symmetric random walk. While the underlying ideas are relatively simple, proving them rigorously requires a number of steps and some new ideas. We start with an outline of the proof.\n\n\\subsection{Outline of proof}\nInstead of considering the expected journey length up to time $t,$ we consider the expected proportion of cars that have parked by time $t.$ It is helpful to restrict ourselves to a finite interval, and this is where car removal strategies become useful. We know by Theorem~\\ref{thm:pinkfloyd} that by removing cars from the process we make it easier for the remaining cars to park. Therefore, any lower bound over an interval for the proportion of unparked cars gives a lower bound for $\\mathbb{E}[\\tau \\wedge t].$\n\nFrom here, we consider a long interval $L \\cup M \\cup R,$ where $L,M,R$ are the left, middle, and right subintervals respectively. We will choose the sizes of $L$ and $R$ so that with high probability no car from $M$ leaves $L \\cup M \\cup R$ by time $t.$ The idea is that with positive probability the number of cars starting in $M$ is a few standard deviations above the mean, creating an excess of cars, and that this excess is not relieved by what happens in $L$ and $R.$ To be able to quantify this, we introduce \\em swapping: \\em this is a way of switching positions of cars so that at any time, from left to right, we see the cars that started in $L,$ then $M,$ and then $R.$ This modification does not change the stochastic properties of the process, but does allow us to say how much relief $L$ and $R$ provide by way of parking spaces available to cars starting in $M.$\n\nFinally, we will bring everything together and appeal to Theorem \\ref{thm:pinkfloyd} to obtain the desired lower bound on $\\mathbb{E}[\\tau \\wedge t].$\n\n\\subsection{The car removal strategy and the swap-modification}\nWe define the car removal strategy $Q$ as follows. Fix integers $k >8$ and $\\ell >4.$ Let $\\zeta = \\lceil \\sqrt{t\\log t} \\rceil$. Then for each integer $r \\in \\mathbb{Z}$ we remove any car which attempts to make a step (in either direction) between $r(2(k+\\ell)\\zeta+1)$ and $r(2(k+\\ell)\\zeta+1)+1$.\n\nWe show that a proportion $(t\\log t)^{-1\/4}$ of cars remains active (i.e., unparked and not removed) at time $t$ under the car removal strategy $Q.$ To establish this, it is sufficient to consider the parking process on an interval of length $2(k+\\ell)\\zeta+1$ where we assume that cars leaving the interval at either end are removed. Let $L = \\mathbb{Z} \\cap [-(k+\\ell)\\zeta,-k\\zeta),$ let $M = \\mathbb{Z} \\cap [-k\\zeta,k\\zeta]$ and $R = \\mathbb{Z} \\cap (k\\zeta,(k+\\ell)\\zeta].$\n\nWe want to show that with positive probability we start with an excess of cars in $M$ which do not escape $L \\cup M \\cup R$ and that $L$ and $R$ do not offer up enough spare parking capacity. It turns out that quantifying what capacity $R$ and $L$ provide is not straightforward since one cannot easily separate what happens to the cars with respect to their starting positions. Particularly problematic is that cars starting in different sections ($L, M$, or $R$) may swap positions. The following modification of the process ensures that at any given time the active cars, as seen from left to right, started their journeys in $L$, then in $M$, and finally in $R$, and will prove very useful.\n\n\\begin{defn}[The modified parking process]\\label{con:modify}\nGiven the parking process $X$, we define a modified process $Y$ as follows. At time $0,$ label cars according to their starting intervals $L,M$ or $R.$ For $s\\geq 0$ we write $C(s)$ for the set of starting positions (in $L \\cup M \\cup R$) of the cars that are still active at time $s$ (hence $C(0)$ is the set of $i$ such that we initially place an active car at $i$). Further we write $C_L(s)$ for the set of starting positions of the cars that started in $L$ and are still active at time $s;$ we similarly define $C_M(s)$ and $C_R(s)$. For a car starting at $i$ which is still active at time $s$ we write $Y^i(s)$ to denote its position at time $s.$\n\nGiven the set $C(s)$ of cars active at time $s,$ and their positions $(Y^i(s): i \\in C(s)),$ we want to define $C(s+1)$ and the positions $(Y^i(s+1) : i \\in C(s+1)).$ We do this in several steps: at each step, we move the cars around in a way that preserves the number of cars at each location. We use $Z^i_1,Z_2^i,$ and $Z_3^i$ to denote intermediate rearrangements, preserving $Y^i$ for the final position. \n\nRoughly speaking: $Z_1$ is where the cars move according to their respective random walks. From $Z_1$ to $Z_2$ we swap cars so that no $L$-car is to the right of an $R$-car. From $Z_2$ to $Z_3$ we swap cars so that no $L$-car is to the right of an $M$-car. Finally, from $Z_3$ to $Y$ we swap cars so that no $M$-car is to the right of an $R$-car. The end result is a swapping of cars which preserves the number of cars at each vertex, is such that cars move by at most one in a single time step, and is such that from left to right the cars have labels $L$, then $M$, and then $R$.\n\n\t\\begin{itemize}\n\t\t\\item For any car active at time $s,$ define $Z^i_1(s+1) = Y^i(s) + (X^i(s+1)-X^i(s)).$\n\t\t\\item Let $i_1, \\ldots, i_{x} \\in L$ (with $Z_1^{i_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $L$ that are active at time $s$ and such that the move at time $s+1$ places them to the right of some active car labelled $R.$ Similarly, let $j_1,\\ldots,j_{y} \\in R$ (with $Z_1^{j_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $R$ that are active at time $s$ and such that the move at time $s+1$ places them to the left of some active car labelled $L.$ \n\n\t\tWe rearrange the cars as follows: for all $i \\notin \\{i_1, \\ldots, i_{x}, j_1,\\ldots,j_{y} \\}$ let $Z^i_2(s+1) = Z^i_1(s+1).$ Let $(m_1, \\ldots, m_{x+y})$ be a permutation of $\\{i_1, \\ldots, i_{x}, j_1, \\ldots, j_{y}\\}$ with $Z^{m_k}_1(s+1)$ increasing in $k$. Then, for $1 \\leq \\ell \\leq x,$ let $Z^{i_\\ell}_2(s+1) = Z^{m_\\ell}_1(s+1),$ and for $1 \\leq \\ell \\leq y,$ let $Z^{j_\\ell}_2(s+1) = Z^{m_{x+\\ell}}_1(s+1).$ After this procedure, no car labelled $L$ is to the right of a car labelled $R.$\t\t\n\t\t\\item Given $Z^i_2(s+1)$ for all $i \\in C(s),$ we define $Z^i_3(s+1)$ by reordering in a similar way the positions $Z^i_2(s+1)$ of the cars that started in $L$ or in $M$ in such a way that no car that started in $L$ has a car that started in $M$ to its left:\n\nLet $i_1, \\ldots, i_{x} \\in L$ (with $Z_2^{i_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $L$ that are active at time $s$ and such that the move at time $s+1$ and the previous rerrangement places them to the right of some active car labelled $M.$ Similarly, let $j_1,\\ldots,j_{y} \\in M$ (with $Z_2^{j_k}(s+1)$ increasing in $k$) be the starting positions of cars labelled $M$ that are active at time $s$ and such that the move at time $s+1$ and previous rearrangement places them to the left of some active car labelled $L.$ \n\n\t\tWe rearrange the cars as follows: for all $i \\notin \\{i_1, \\ldots, i_{x}, j_1,\\ldots,j_{y} \\}$ let $Z^i_3(s+1) = Z^i_2(s+1).$ Let $(m_1, \\ldots, m_{x+y})$ be a permutation of $\\{i_1, \\ldots, i_{x}, j_1, \\ldots, j_{y}\\}$ with $Z^{m_k}_2(s+1)$ increasing in $k$. Then, for $1 \\leq \\ell \\leq x,$ let $Z^{i_\\ell}_3(s+1) = Z^{m_\\ell}_2(s+1),$ and for $1 \\leq \\ell \\leq y,$ let $Z^{j_\\ell}_3(s+1) = Z^{m_{x+\\ell}}_2(s+1).$\n\nNote that this operation can only move cars labelled $L$ to the left; hence we still have no car labelled $L$ to the right of a car labelled $R.$\n\t\t\\item Finally, given $Z^i_3(s+1)$ for all $i \\in C(s),$ we define $Y^i(s+1)$ by reordering in a similar way the positions $Z^i_3(s+1)$ of the cars that started in $M$ or in $R$ in such a way that no car that started in $R$ has a car that started in $M$ to its right. Again, note that this operation only moves  cars labelled $R$ to the right, hence we still have no car labelled $L$ to the right of a car labelled $R.$ Moreover, a car labelled $M$ can only be moved to a position $Z^i_3$ previously occupied by a car labelled $M$ or $R,$ which we know has no car labelled $L$ to its right; hence the same holds about cars labelled $M$ after the rearrangement.\n\t\\end{itemize}\n\nIf a single car starting at $i$ reaches an empty parking space at $Y^i(t),$ then it parks there. When at least two cars simultaneously arrive at a parking space $v$ at time $t,$ we choose the car $i$ labelled $L$ with smallest $U^i_t$ to park there; in the absence of a car labelled $L,$ the car $i$ labelled $R$ with smallest $U^i_t$ parks there; finally, if only cars labelled $M$ meet at $v,$ the car $i$ with smallest $U^i_t$ parks there. When a car leaves $L \\cup M \\cup R,$ we say it is \\em inactive \\em and remove it from the process. We say that a car becomes \\emph{left-inactive} if it reaches $\\min L -1,$ and it becomes \\emph{right-inactive} if it reaches $\\max R +1.$ Finally, let $C(s+1) \\subseteq C(s)$ be the set of cars active at time $s$ that have neither parked nor become inactive at time $s+1.$\n\\end{defn}\n\n\\begin{rem}\n \\label{rem:limitedDrive}\nIn the process described in Definition \\ref{con:modify}, for any $i \\in \\mathbb{Z},$ $Y^i(s+1) - Y^i(s) \\in \\{-1,0,+1\\}$ -- the total move of a car in a step is at most one. Indeed, consider an arbitrary car $i$ labelled $M$ with $Y^i(s) = j$. At time $s$ it has no cars labelled $L$ strictly to its right and no cars labelled $R$ strictly to its left. At time $s+1,$ all cars labelled $L$ can only drive to positions at most $j+1$ (so $Z^k_1(s+1) \\le j+1$ for each $k \\in C_L(s)$), and cars labelled $R$ drive to positions at least $j-1$ (so $Z^k_1(s+1) \\ge j-1$ for each $k \\in C_R(s)$). It is not possible to move $i$ to a position strictly to the left of the left-most (according to $Z_1$) car labelled $R$ so that $Y^i(s) \\ge j-1$. Similarly it is not possible to move $i$ to a position strictly to the right of the right-most (according to $Z_1$) car labelled $L$ so that $Y^i(s) \\leq j+1$. Similar arguments apply to cars labelled $L$ or $R.$\n\\end{rem}\n\nLet $\\widetilde\\mathbb{P}$ be the probability measure with respect to the modified parking process. If we ignore the labels of the cars, then the difference from the original parking process under $Q$ is that we swap some future trajectories of cars. Since the swapping is determined by past trajectories, the unlabelled modified process has the same distribution as the original parking process with car removal strategy $Q$. Thus \n\t\\begin{align}\n\t\t\\widetilde\\mathbb{E}[\\#\\mbox{active cars in $L \\cup M \\cup R$ at time $t$}] = \\mathbb{E}^Q[\\#\\mbox{active cars in $L \\cup M \\cup R$ at time $t$}]. \\label{telematch}\n\t\\end{align}\n\n\\subsection{Proof of the lower bound}\nBefore completing the proof of Theorem \\ref{thm:p=1\/2}, we prove some preliminary lemmas concerning the modified parking process. Unless stated otherwise, we assume that we are dealing with the modified parking process (Definition \\ref{con:modify}) throughout this section.\n\nFirst we consider how many cars from $L$ and $R$ become inactive. Intuitively this should be maximised if the cars drive monotonically towards the ends of the interval. Given the initial arrangement of cars and parking spaces on $L,$ let $D_L = D_L(t)$ be the number of cars starting in $L$ which would become left-inactive by time $t$ should all cars with label $L$ move left deterministically. Similarly let $D_R = D_R(t)$ denote the number of cars with label $R$ that become right-inactive by time $t$ in the process where all cars with label $R$ move right deterministically. The next lemma shows that this intuition is correct.\n\n\\begin{lem}\\label{claim:lemming}\nThe number of cars with label $L$ which become left-inactive by time $t$ is at most $D_L.$\n\\end{lem}\n\n\\begin{proof}\nUnder $\\widetilde\\mathbb{P},$ suppose that $j$ is the smallest integer which has a parking space either unfilled or filled by a car labelled $M$ or $R$ by time $t.$ Let $J = L \\cap [-(k+\\ell)\\zeta,j-1].$ We claim that the only cars labelled $L$ that can become left-inactive are the cars from $J,$ and only cars originating in $J$ park in $J.$ First suppose that $j$ is unfilled. No car from the right of $j$ passes through $j$ (else it would park there) and so no car from the right of $j$ can become left-inactive.\n\nSo suppose that car $w$ (labelled $M$ or $R$) parks in $j$. Under $\\widetilde\\mathbb{P}$ at any time, from left to right, the unparked cars have labels $L,$ then $M,$ and then $R.$ Therefore, any car $v$ labelled $L$ originating from an integer greater than $j,$ before it parks, must stay to the left of the car $w$ which parks in $j.$ Since cars in the modified process move at most one step at each time, the car $v$ cannot be unparked at time $t$ since it would have visited $j$ before $w$ parks there. Similarly, $v$ cannot park to the left of $j$ since it would first pass through $j$ (before $w$ parks there). Therefore, any car labelled $L$ originating from an integer greater than $j$ must have parked in a spot greater than $j.$\n\nSuppose that cars starting at positions $i_1<\\dots<i_N < j$ become left-inactive starting from $J.$ Observe that every parking space to the left of $i_N$ must be filled by a car originating from $J$ (otherwise, the car starting in $i_N$ must reach a free parking space on its route to $\\min L -1$). Let $p=j-1$ if all parking places in $J$ are filled in the process, and otherwise let $p+1$ be the leftmost empty parking space in $J$ at time $t.$ We see that all parking spaces to the left of $p+1$ must be filled by cars originating from the left of $p+1$ (a car starting to the right of $p$ would fill $p+1$ first). But then there must be a surplus of $N$ cars to the left of $p+1.$\n\nIf all the cars drove left deterministically, this surplus would result in at least $N$ cars, starting to the left of $p+1,$ becoming left-inactive. Thus we have $D_L \\geq N,$ proving the claim.\n\\end{proof}\n\n\\begin{rem}\nClearly, the analogous claim that the number of cars from $R$ becoming right-inactive is bounded by $D_R$, also holds.\n\\end{rem}\n\nNote that $D_L$ and $D_R$ are dependent only on the initial car configuration $(B^i)_{i \\in \\mathbb{Z}}.$ Let $S_L$ be the number of cars which start in $L$ and let $P_L$ be the number of parking spaces in $L$ (hence clearly $S_L+P_L = |L|$). Similarly define $S_R$ and $P_R.$\n\n\\begin{lem}\\label{claim:easing}\nThere exists $\\varepsilon > 0$ (independent of $t$) such that\n\\[\n \\mathbb{P}\\left[S_L-P_L-D_L \\ge -(t\\log t)^{1\/4}\\right] > \\varepsilon.\n\\]\n\\end{lem}\n\n\\begin{proof}\nConsider the simple symmetric random walk starting at $0$ which increases at time $i \\geq 1$ if the $i$th rightmost point in $L$ initially contains a car, and decreases if the $i$th rightmost point in $L$ contains a parking space. Suppose that while traversing $L,$ the walk last attains its minimum value $-m \\leq 0$ at time $j,$ and let $x$ be the $j$th rightmost point in $L.$ Then, in the process where all cars in $L$ deterministically drive left, every car starting to the right of $x$ finds a parking place, the process ends with $m$ empty spots to the right of $x-1,$ every spot to the left of $x$ is filled by a car, and all the cars that do not park reach the left end of the interval and become left-inactive.\n\nThe number of parked cars in this process is $S_L-D_L,$ and so the number of unfilled parking spaces is $P_L-S_L+D_L.$ Therefore $S_L - P_L-D_L = -m.$ From the previous paragraph, we see that $S_L - P_L-D_L$ is distributed like the minimum of a simple symmetric random walk of length $\\ell \\zeta.$ So by Lemma \\ref{minssrw}(ii) it is at least $-(t\\log t)^{1\/4}$ with probability bounded away from zero.\n\\end{proof}\n\\begin{rem}\n An analogous claim holds if we replace $S_L, P_L, D_L$ with $S_R, P_R, D_R$ respectively.\n\\end{rem}\n\nWe would like to say that no car from $L$ becomes right-inactive. Indeed, we could then say that at time $t,$ the number of cars from $L$ (possibly parked) still in $L\\cup M \\cup R$ minus the number of parking spaces (filled or unfilled) in $L$ is at least $S_L-P_L-D_L \\ge -(t\\log t)^{1\/4}$ with probability at least $\\varepsilon.$ The next result shows that this occurs, and also that no car from $M$ becomes inactive.\n\n\\begin{lem}\\label{claim:restrict}\nWith probability $1-o(1\/t),$ the random walks $\\left(X^i\\right)_{i\\in L \\cup M \\cup R}$ are such that for all possible starting configurations of active cars and parking places in $L \\cup M \\cup R,$ in the first $t$ time steps: no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$\n\\end{lem}\n\\begin{proof}\nFor each $i \\in L \\cup M \\cup R,$ let $M^i$ be the maximum of $\\{X^i_s - i : s \\le t\\},$ and $m^i$ the minimum of $\\{X^i_s -i : s \\le t\\}.$ By Lemma \\ref{minssrw} (i), $\\mathbb{P}\\left[m^i \\le - 4\\sqrt{t \\log t}\\right]=\\mathbb{P}\\left[M^i \\ge 4\\sqrt{t \\log t}\\right] \\le 2t^{-8}.$ Hence by the union bound, with failure probability $o(t^{-1}),$ for all $i \\in L \\cup M \\cup R$ the random walks $X^i$ are at distance at most $4\\zeta$ from their corresponding starting point $i$ until time $t.$\n\nAssume that for all $i \\in L \\cup M \\cup R,$ $X^i$ is at distance at most $4\\zeta$ from $i$ until time $t.$ We now show that for all starting configurations of active cars and parking places in $L \\cup M \\cup R,$ in the first $t$ time steps, no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$\n\nConsider a car starting at $i \\in L.$ If the car is still active at time $s$ in the modified parking process, then $Y^i(s) \\leq X^i(s),$ as if the position of the car is ever changed as a result of landing to the right of a car labelled $M$ or $R,$ then it can only be pushed further left. Therefore it stays to the left of $(4-k)\\zeta.$ Similarly all cars labelled $R$ stay to the right of $(k-4)\\zeta.$ Since $k > 8,$ no car from $L$ reaches $R,$ and vice versa.\n\nNow consider a car starting at $i \\in M.$ If the position of the car is never changed due to moving past a car labelled $L$ or $R,$ then it never reaches a point more than $4\\zeta$ from $i$ and so cannot become inactive (recall that $\\ell > 4$).\n\nHence suppose the car at some point has its position changed due to finding itself to the left of a car labelled $L.$ This implies that the car must at some point be to the left of $(4-k)\\zeta$ (or else it cannot pass a car labelled $L$). If the car reaches $(k-4)\\zeta + 1$ at some point, then there must be a passage of the car between $(4-k)\\zeta$ and $(k-4)\\zeta$ contained within $[(4-k)\\zeta,(k-4)\\zeta].$ In this segment, the position of the car cannot be changed as it keeps all cars labelled $L$ to its left, and all cars labelled $R$ to its right. Therefore it moves according to $X^i,$ and so $X^i$ reaches points $2(k-4)\\zeta > 8 \\zeta$ apart (recall that $k > 8$). This cannot happen since the maximum modulus of $X^i-i$ is at most $4\\zeta.$\n\nTherefore the car does not have its position changed due to being to the right of a car labelled $R.$ So while the car remains active, its position is bounded below by $X^i$ (having its position changed can only push its the car to the right). Since the car does not reach $(k-4)\\zeta,$ we see that the position of the car is contained in $[(-k-4)\\zeta,(k-4)\\zeta]$ and so the car cannot become inactive (as $\\ell > 4$).\n\nThe argument for a car which at some point finds itself to the right of a car labelled $R$ is identical. We conclude that no car originating from $M$ becomes inactive.\n\\end{proof}\n\nWe are now in a position to prove Theorem \\ref{thm:p=1\/2}.\n\n\\begin{proof}[Proof of the lower bound in Theorem \\ref{thm:p=1\/2}]\nIt is enough to show that with probability bounded away from zero (say at least $\\delta > 0$), at time $t$ there are at least $(t \\log t)^{1\/4}$ active cars in $L \\cup M \\cup R$ in the modified process. If this holds, then the result easily follows by symmetry, Theorem \\ref{thm:pinkfloyd} and \\eqref{telematch}:\n\t\\begin{align*}\n\t\t\\mathbb{E}[\\tau \\wedge t] & = \\frac{\\mathbb{E}^N\\left[\\sum_{v \\in L\\cup M \\cup R} \\tau^v \\wedge t\\right]}{|L \\cup M \\cup R|} \\\\\n\t\t& \\ge \\frac{\\mathbb{E}^Q\\left[\\sum_{v \\in L\\cup M \\cup R} \\tau^v \\wedge t\\right]}{|L \\cup M \\cup R|} \\\\\n\t\t& \\ge \\frac{\\widetilde\\mathbb{E}[\\# \\mbox{active cars at time $t$ in $L \\cup M \\cup R$}]}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& = \\frac{\\widetilde\\mathbb{E}[\\# \\mbox{active cars at time $t$ in $L \\cup M \\cup R$}]}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& \\ge \\frac{\\delta (t \\log t)^{1\/4}}{2(k+\\ell)\\zeta+1} \\cdot t  \\\\\n\t\t& = \\Omega(t^{3\/4} \\log^{-1\/4} t).\n\t\\end{align*}\n\nLet $I_L$ be the number of cars starting in $L$ that become left-inactive and let $I_R$ be the number of cars starting in $R$ that become right-inactive. Analogously to $L$ and $R$, let $S_M$ be the number of cars which start in $M$ and let $P_M$ be the number of initial parking places in $M.$ Hence, in total there are $P_L+P_M+P_R$ parking places in $L \\cup M \\cup R.$\n\nSuppose that in the first $t$ steps of the process, no car starting in $M$ becomes inactive, no car starting in $L$ reaches $R,$ and no car starting in $R$ reaches $L.$ Then at time $t,$ the number of cars (active or parked) in $L \\cup M \\cup R$ is $S_M  + (S_L - I_L) + (S_R  - I_R).$ By Lemma \\ref{claim:lemming} this is at least $S_M  + (S_L - D_L) + (S_R  - D_R).$ Since only one car can park in a parking space, the number of active cars in $L \\cup M \\cup R$ at time $n$ must be at least\n\t\\begin{eqnarray}\n\t\t(S_M-P_M) + (S_L-P_L)-D_L + (S_R-P_R)-D_R. \\label{eq:nexcess}\n\t\\end{eqnarray}\n\nObserve that $S_M-P_M$ is determined by the starting configuration in $M,$ $S_L-P_L-D_L$ is determined by the starting configuration in $L,$ and $S_R-P_R-D_R$ is determined by the starting configuration in $R.$ Therefore these random variables are mutually independent. Let $C^M$ be the event that $S_M - P_M$ is at least $3(t\\log t)^{1\/4},$ let $C^L$ be the event that $S_L-P_L-D_L \\geq -(t\\log t)^{1\/4},$ and let $C^R$ be the event that $S_R-P_R-D_R \\geq - (t\\log t)^{1\/4}.$\n\nLet $A$ be the random event, depending on the random walks $X^i$ only, that for all initial configurations of cars and parking places in $L \\cup M \\cup R,$ no car from $M$ becomes inactive, no car from $L$ reaches $R,$ and no car from $R$ reaches $L.$ Observe that $A,C^L, C^M,C^R$ are mutually independent events. By Lemma \\ref{claim:restrict}, $A$ occurs with high probability. By Lemma \\ref{claim:easing} both $C^L$ and $C^R$ occur with probability bounded away from zero. Let $K = 2k\\zeta + 1 \\approx 2k\\sqrt{t\\log t}.$ Since $$S_M - P_M = 2k\\zeta +1 - 2P_M \\sim K - 2\\mathrm{Bin}(K,1\/2),$$ we have\n\t\\begin{align*}\n\t\t\\mathbb{P}\\left[C^M\\right] & = \\mathbb{P}\\left[\\mathrm{Bin}(K,1\/2) \\leq \\frac{K - 3(t\\log t)^{1\/4}}{2}\\right]  \\\\\n\t\t& = \\mathbb{P}\\left[\\frac{\\mathrm{Bin}(K,1\/2) - \\frac{K}{2}}{\\sqrt{\\frac{K}{4}}} \\leq \\frac{-6(t\\log t)^{1\/4}}{\\sqrt{K}}\\right].\n\t\\end{align*}\nBy the Central Limit Theorem, this probability tends to $\\Phi(-\\frac{6}{\\sqrt{2k}})$ as $t$ tends to infinity. Therefore $C^M$ occurs with probability bounded away from zero. So all four events $A,C^L, C^M,C^R$ occur simultaneously with probability bounded away from zero.\n\nSuppose that the events $A,C^L,C^M,C^R$ all occur. Then recalling equation \\eqref{eq:nexcess} we see that the number of active cars in $L \\cup M \\cup R$ at time $t$ is at least\n\\begin{align*}\n (S_M-P_M) + (S_R-P_R)-D_R + (S_L-P_L)-D_L & \\ge 3 (t\\log t)^{1\/4} - (t\\log t)^{1\/4} - (t\\log t)^{1\/4} \\\\\n  & = (t\\log t)^{1\/4}.\n\\end{align*}\n\nWe conclude that with probability bounded away from zero, there are at least $(t\\log t)^{1\/4}$ active cars in $L \\cup M \\cup R$ at time $t.$ The lower bound $\\Omega(t^{3\/4} \\log^{-1\/4} t)$ on $\\mathbb{E}[\\tau \\wedge t]$ follows.\n\\end{proof}\n\n\\section{Subcritical parking on $\\mathbb{Z}$}\n\\label{sec:p<1\/2}\n\nIn this section we prove Theorems \\ref{thm:p<1\/2} and \\ref{thm:critexp}. This is done in two parts. First, for a car starting at $0$, we consider the smallest $J$ (depending only on the initial configuration of cars) such that no matter what the other cars do, there is always a free parking space in both $[1,J]$ and $[-J,-1]$. Given $J$, we know that the car starting at $0$ cannot reach either $-J$ or $J$ before it parks. Calculating the expected journey length of $0$ is then carried out by proving tail bounds on the random variable $J$.\n\nWe start with the following simple lemma, which we state here without proof.\n\n\\begin{lem}\\label{lem:markstat}\nLet $p \\in (0,1\/2),$ and let $Y=Y(p)$ be a Markov chain on $\\mathbb{N} \\cup \\{0\\}$ with $Y_0=0$ and transition probabilities $(p_{i,j})_{i,j \\in \\mathbb{N} \\cup \\{0\\}}$ where\n\\[\np_{i,j} = \\begin{cases}\np, \\quad & j = i+1, \\\\\n1-p, \\quad & j = i - 1 \\ge 0 \\mbox{ or } i=j=0, \\\\\n0, \\quad & \\mbox {otherwise.}\n\\end{cases}\n\\]\nThen $Y$ has stationary distribution $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$. Furthermore, since $Y$ is an aperiodic and irreducible Markov chain, $Y_t \\rightarrow \\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ in distribution as $t \\rightarrow \\infty$.\n\\end{lem}\n\nLet $E^L(t)$ be the number of cars in $[-t,-1]$ that would reach $0$ if all cars deterministically drove right. We also define $E^R(t)$ to be the number of cars in $[1,t]$ which would reach $0$ if all cars deterministically drove left. Note that $(E^L(t))_{t \\in \\mathbb{N}}$ is an increasing sequence of random variables. Finally, let $E^L$ be the number of cars in $(-\\infty,-1]$ that would reach $0$ if all cars deterministically drove right (and analogously define $E^R$). Note that $E^L(t)$ increases almost surely to $E^L$ as $t \\rightarrow \\infty$.\n\n\\begin{lem}\n\\label{lem:thedescent}\nFor all $p < 1\/2,$ $E^L \\sim \\Geom_{\\ge 0}(\\tfrac{1-2p}{1-p})$.\n\\end{lem}\n\\begin{proof}\nSince $(E^L(t))_{t \\ge 1}$ increases almost surely to $E^L$, it is sufficient to show that $E^L(t) \\to \\Geom_{\\ge 0}(\\tfrac{1-2p}{1-p})$ in distribution as $t \\to \\infty$. To compute $E^L(t)$, consider forming a queue of cars from left to right in $[-t,-1]$: Let $Q_0 = 0$ (there is initially no queue), then given $Q_i$, we set $Q_{i+1} = Q_i+1$ if there is initially a car at $i-t$ (a car is added to the queue), $Q_{i+1} = Q_i-1$ if $Q_i > 0$ and there is initially a parking space at position $i-t$ (a car from the queue is parked), and $Q_{i+1}=0$ otherwise. Then $Q_t = E^L(t)$. On the other hand, $(Q_s : s \\le t)$ is distributed like $(Y_s : s \\le t)$ in Lemma \\ref{lem:markstat}, and so $E^L(t)$ has the same distribution as $Y_t$ (with $Y_0 = 0$). By Lemma~\\ref{lem:markstat}, $E^L(t) \\rightarrow \\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ in distribution as $t \\rightarrow \\infty$.\n\\end{proof}\n\nClearly, $E^R(t)$ also increases almost surely to the random variable $E^R$ which is distributed like a $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ random variable (and is independent of $E^L$).\n\nFor all $r \\ge 0$, let ${E}_r^R(t)$ be the number of cars in $[r+1,r+t]$ that would reach $r$ if all cars deterministically drove left, and similarly let ${E}_r^L(t)$ be the number of cars in $[-r-t,-r-1]$ that would reach $-r$ if all cars deterministically drove right. Let ${E}_r^R$ and ${E}_r^L$ be the limits as $t \\to \\infty$ respectively of ${E}_r^R(t)$ and ${E}_r^L(t)$. Note that Lemma \\ref{lem:thedescent} holds with $E^L$ replaced by $E^R_r$, as well as by $E^L_r$. For all $r \\ge 1$, let $S_r^R$ and $S_r^L$ be the number of cars that start in $[1,r]$ and $[-r,-1]$ respectively.\n\nIn the proof of Theorem \\ref{thm:p<1\/2}, we show that at most ${E}_K^R+E^L + S_K^R$ cars from $\\mathbb{Z}\\setminus \\{0\\}$ can be present in $[1,K]$ at any time. This means that at most ${E}_K^R+E^L + S_K^R$ parking spaces in $[1,K]$ can be filled by cars from $\\mathbb{Z} \\setminus \\{0\\}$. Therefore, if ${E}_K^R+E^L + S_K^R < K\/2,$ then there must be a parking space in $[1,K]$ not filled by a car from $\\mathbb{Z} \\setminus \\{0\\}$ (consider that there are initially $K-S_K^R$ parking spaces in $[1,K]$). It follows that a car starting at $0$ parks before reaching $K$.\n\nIn the proof of Theorem \\ref{thm:p<1\/2}, we first condition on the smallest $K$ such that both ${E}_K^R+E^L + S_K^R < K\/2$ and ${E}_{K}^L+E^R + S_K^L < K\/2.$ These conditions mean that a car starting at $0$ will have parked by the time its associated random walk $X^i$ hits either $-K$ or $K.$\n\n\\begin{proof}[Proof of Theorem \\ref{thm:p<1\/2}]\nLet $p < 1\/2$ and let $J$ be the smallest $K$ such that ${E}_K^R+E^L + S_K^R < K\/2$ and ${E}_{K}^L+E^R + S_K^L < K\/2$ if such a $K$ exists, and let $J = \\infty$ otherwise. For a given $a \\in \\mathbb{N},$ let $H_a$ be the first hitting time of $a$ by the random walk $X^0.$ We claim that if $J=N,$ then $\\tau^0 \\le H_{-N} \\wedge H_N.$ We justify this by showing that at any time $t \\ge 0,$ there are at most ${E}_N^R+E^L + S_N^R$ cars excluding car $0$ (parked or not) present in $[1,N]$ at time $t.$ A similar statement can be shown for cars present in $[-N,-1].$\n\nLet us temporarily exclude the car starting at $0$ from the parking process (e.g., assume that this car never decides to park) and suppose that at time $t,$ there are $B$ cars that started in $[N-t+1,N]\\setminus \\{0\\}$ parked in $[N+1,N+t].$ Let $R$ be the number of cars that start in $[N+1,N+t]$ that are in $[N-t+1,N]$ at time $t.$ By an argument identical to that of Lemma \\ref{claim:lemming}, we have the bound $R \\le B + E_N^R(t)$ since each parked car from $[N-t+1,N]$ that parks inside $[N+1,N+t]$ can only increase the number of cars that reach $N$ from $[N+1,N+t]$ by $1.$ Similarly, if $C$ is the number of cars that started in $[1,t]$ and parked in $[-t,-1],$ and $L$ is the number of cars that start in $[-t,-1]$ present in $[1,N]$ at time $t,$ we have $L \\le C + E^L(t).$ So the number of cars present in $[1,N]$ at time $t$ is\n\t\\begin{align*}\n\t\tS_N^R + R + L - B - C & \\leq S_N^R + E^L(t) + E_N^R(t) \\\\\n\t\t& \\leq S_N^R + E^L + E^R_N.\n\t\\end{align*}\nSince $J = N,$ this quantity is strictly less than $N\/2.$ On the other hand, there are initially $N-S_N^R > N\/2$\nparking spaces in $[1,N]$ and so car $0$ must go through an empty parking space before reaching $N.$ A similar argument applies to $[-N,-1].$ In the real process, where car $0$ tries to park, this implies car $0$ parks before reaching $N$ or $-N.$\n\nIf $J < \\infty$ almost surely, we therefore have\n\t\\[\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}\\mathbb{P}\\left[J=N\\right]\\mathbb{E}[H_{-N} \\wedge H_N | J = N].\n\t\\]\nBy independence and Lemma \\ref{lbeforer} (iii) we have\n\\[\n\t\\mathbb{E}[H_{-N} \\wedge H_N | J = N] = \\mathbb{E}[H_{-N} \\wedge H_N] = N^2,\n\\]\nand so, assuming again that $J < \\infty$ with probability $1$,\n\t\\begin{align}\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}N^2 \\mathbb{P}\\left[J=N\\right]. \\label{Qbound}\n\t\\end{align}\n\nWe now consider the distribution of $J$. If $J$ is at least $N,$ then by averaging one of the following must happen:\n\\begin{itemize}\n\\item[(i)] One of $S_N^R$ and $S_N^L$ is at least $N(p + (1\/4 - p\/2)).$\n\\item[(ii)] One of $E^L, E^R, {E}_{N}^L$, and ${E}_N^R$ is at least $N(1\/8-p\/4).$\n\\end{itemize}\nClearly $S_N^R$ and $S_N^L$ are both distributed like $\\Bin(N,p)$ random variables and so, by Lemma \\ref{feelthechern}, the probability that (i) occurs is at most $2e^{-(1\/2-p)^2N\/2}.$ On the other hand, by Lemma \\ref{lem:thedescent}, we know that $E^L, E^R, {E}_N^L$ and ${E}_{N}^R$ are all distributed like $\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$ random variables. If $X~\\sim~\\Geom_{\\ge 0}(\\frac{1-2p}{1-p})$, then $\\mathbb{P}\\left[X \\geq N(1\/8-p\/4)\\right] \\leq \\bigl(1-\\frac{1-2p}{1-p}\\bigr)^{N(1\/8-p\/4)}$, and so the probability that (ii) occurs is at most\n\\[\n4\\mathbb{P}\\left[\\Geom_{\\ge 0}\\left(\\frac{1-2p}{1-p} \\right) \\ge N(1\/8-p\/4)\\right] \\leq 4\\left(1-\\frac{1-2p}{1-p}\\right)^{N(1\/8-p\/4)}.\n\\]\nPutting these together we see that for all $N \\ge 1$ we have\n\\begin{align*} \n\t\t\\mathbb{P}\\left[J = N\\right] \\le \\mathbb{P}\\left[J \\ge N\\right] & \\le 2e^{-(1\/2-p)^2N\/2} + 4\\biggl(1-\\frac{1-2p}{1-p}\\biggr)^{N(1\/8-p\/4)} \\\\\n\t\t& \\le 2e^{-(1\/2-p)^2N\/2} + 4e^{-\\frac{1-2p}{1-p}N(1\/8-p\/4)}.\n\\end{align*}\nAs the above bound on $\\mathbb{P}\\left[J \\ge N\\right]$ tends to $0$ as $N \\to \\infty$, we see that $J < \\infty$ almost surely. Hence, putting the obtained bound into \\eqref{Qbound} gives\n\t\\begin{align}\n\t\t\\mathbb{E}[\\tau^0] \\le \\sum_{N \\ge 1}N^2\\left[2e^{-(1\/2-p)^2N\/2} + 4e^{-\\frac{1-2p}{1-p}N(1\/8-p\/4)}\\right]. \\label{critequat}\n\t\\end{align}\nThis series converges for any $p < 1\/2$ and so $\\mathbb{E}[\\tau^0]$ is finite.\n\\end{proof}\n\nFinally, we come to proving Theorem \\ref{thm:critexp}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:critexp}]\nLet $p < 1\/2$ and write $\\varepsilon = 1\/2 - p$. By \\eqref{critequat} \n\n\t\\begin{align*}\n\t\t\\mathbb{E}[\\tau^0] &\\le \\sum_{N \\ge 1}N^2\\left[2e^{-\\frac{\\varepsilon^2}{2}N} + 4e^{-\\frac{\\varepsilon^2}{1+2\\varepsilon}N}\\right]  \\le 6\\sum_{N \\ge 1} N^2e^{-\\varepsilon^2N\/2}.\n\t\\end{align*}\n\nThis sum can be approximated by the integral $\\int_{0}^{\\infty}x^2e^{-\\varepsilon^2x\/2}dx$. By considering the pdf of a $\\Gamma\\left(3,\\varepsilon^2\/2\\right)$ random variable we get a bound of the form $O\\left(\\varepsilon^{-6}\\right).$\n\n\\end{proof}\n\n\\section{Further questions}\\label{yaymorewaffle}\nThere is still a gap between the upper and lower bounds in Theorem \\ref{thm:p=1\/2}. Following the conjecture presented in \\cite{Junge-seminar}, we also believe that the upper bound gives the right order $t^{3\/4}.$\n\nIt would be interesting to know what happens in higher dimensions, where the problems seem to become more difficult and are likely to require additional ideas. It is also natural to ask what happens in other lattices: for example, are there analogous results to Theorems \\ref{thm:p=1\/2} and \\ref{thm:p<1\/2} that hold for the hexagonal lattice? We remark that Damron, Gravner, Junge, Lyu and Sivakoff \\cite[Open Questions 1 and 2]{DGJLS} have conjectures here (which we believe to be true).\n\nFinally, what can we say for more general jump distributions? We conjecture that if the increments of the random walks $X^i$ on $\\mathbb{Z}$ are bounded, then Theorems \\ref{thm:p=1\/2} and \\ref{thm:p<1\/2} should still hold. Although similar methods could work, one would have to be careful about specifying parking places for cars (as in the parking strategy $T$ in Section~\\ref{sec:p=1\/2upper}) as cars might jump over them.\n\n\\bibliographystyle{amsplain}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Hawking radiance of a black hole is usually characterized by the mean number of quanta emitted in each mode~\\cite{Hawking0}.  A more precise characterization entails specification of the probability distribution of the number of quanta in each mode.  Such distributions were first calculated early~\\cite{Parker,Wald,Hawking}.  In later years the issue of unitary evolution has convinced many workers that the radiance must be in a pure quantum state.  Full characterization of such state would entail, at the very least, giving the \\emph{joint} probability distribution of the number of quanta emitted in the various modes.  Since the purity of the radiance raises still unsolved problems, we focus here on the one-mode marginal probability distributions; by marginalizing the joint distribution we remain neutral with regard to the ultimate outcome of the said controversy.  \n\nMore specifically, we here investigate the relation between the one-mode response of a Kerr black hole to incident bosonic radiation and the energy spectrum for Kerr black holes.  The question we focus on is whether any insight into quantum gravity can be had from such investigation. The ready answer would be that the thermality of the Hawking radiance, which seems incontestable if the purity status is rejected, should make it blind to any specifics of the black hole structure and dynamics, so that nothing can be learned from it about quantum gravity.  In our view this is a simplistic response.  To draw an analogy, the radiation emitted by an atom says a lot about its quantum structure. Only when numerous atoms radiate together incoherently does the radiation assume a thermal guise. Now to some extent the black hole, in its simplicity, is the hydrogen atom of quantum gravity~\\cite{BekMG,Corda}.  By analogy we could expect to glean insights into this subject from detailed analysis of the black hole radiance which, after all, emanates from a coherent gravitational structure, not from an assembly of quantum systems.  Indeed, as we shall show, a definite assumption about the horizon area spectrum of a black hole recovers, by simple argumentation, the correct one-mode response probability distribution.\n\nWe shall focus here on the model of a quantum black hole espoused in Refs.~\\cite{spectrum,Mukh,BekMukh}.   Its central claim is that the horizon area $A$ (or entropy $S$) is quantized in equally spaced steps.  For the Kerr black hole (mass $M$ and angular momentum $J$) this means (we take $c=1$)\n\\begin{equation}\nA=4\\pi G^2 \\big(M+\\sqrt{M^2-J^2\/G}\\big)^2 =\\alpha L_P{}^2\\, n;\\quad  n=1,2,\\cdots,\n\\label{Aq}\n\\end{equation}\nwhere $\\alpha$ is a dimensionless positive constant.  Two possible values of $\\alpha$ are often mentioned: $\\alpha=8\\pi$~\\cite{spectrum,Maggiore} and $\\alpha=4 \\log k$ with $k=2,3,\\cdots$~\\cite{Mukh,BekMukh,Hod}.  We shall here focus on macroscopic holes, those with very large $n$. We assume that the fundamental transition of a black hole with consequent emission or absorption of one radiation quantum is one with $n\\Rightarrow n\\mp1$.  Conservation laws then fix the frequency and azimuthal quantum number of the quanta; for emission $\\hbar\\omega=-\\Delta M$ and $\\hbar\\tilde m=-\\Delta J$ with $\\tilde m\\in (\\cdots,-2,-1,0,1,2,\\cdots)$.  Thus differencing (more practically differentiating) \\Eq{Aq} gives as the characteristic parameter for the fundamental transition\n\\begin{equation}\nx\\equiv \\hbar(\\omega-\\tilde m\\Omega)\/T_{\\rm H} =\\alpha,\n\\label{x}\n\\end{equation}\nwhere $T_{\\rm H}$ and $\\Omega$ are the Hawking temperature and rotational angular frequency for the Kerr black hole.  [Use has been made of the area formula, $S_{\\rm BH}=A(4L_P{}^2)^{-1}$, as well as the thermodynamic relations $(\\partial S_{\\rm BH}\/\\partial M)_J=1\/T_{\\rm H}$ and $(\\partial S_{\\rm BH}\/\\partial J)_M=-\\Omega\/T_{\\rm H}$].  It may be seen that the fundamental transition's $x$ parameter, or $\\omega$ for fixed $\\tilde m$, is constant over a moderate range of $M$ or $A$.  This means that a moderately long series of quanta with like values of $\\omega$ and $\\tilde m$ can be emitted by a macroscopic black hole making successive $n\\Rightarrow n-1$ transitions before the frequency begins to shift and a switch to another mode can occur.  Likewise, transitions with $n\\Rightarrow n-2, n\\Rightarrow n-3, \\cdots$ can also take place, with values of the parameter $x$ which are integral multiples of that in \\Eq{x}.  These too can occur in series.\nEvery such series gives rise to a spectral line; there are many spectral lines for $n\\Rightarrow n-1$ (likewise for $n\\Rightarrow n-2$, etc.), each for a different value of $\\tilde m$.\n\nWe remark that any other quantization rule according to which $A$ is a smooth function of a quantum number analogous to $n$ will lead to similar conclusions regarding the allowed values of $\\omega$.  Not so when a black hole parameter different from $A$ is subject to quantization. Likewise, completely different results would be expected for a continuum horizon area spectrum.\n\n\n\\section{Jump probabilities}\n\\label{sec:jump}\n\nFrom now on we refer to transitions between $A$ levels as jumps.  Let us focus on two adjacent area levels of an isolated black hole, $u$ and $l$.  Each quantum state belonging to the upper level $u$ has a definite probability to jump-down to a state belonging to the lower level $l$ whereby a quantum with a definite $\\tilde m$ is emitted; we denote this \\emph{elementary probability} by $e^{-\\beta}$ with $\\beta$ real and positive~\\cite{Fulfill}; to reduce clutter we do not the $x$ or $\\tilde m$ of the mode.  Regarding the time span to which this probability refers, we shall take it as the time associated with emission of the mode of black hole radiation which emerges from the said transition.  Of course the time depends on how monochromatic we take the mode to be; very monochromatic modes are emitted over a long time.    Once the black hole has jumped down one area level, it is in a situation similar to the initial one, so another jump to one level lower should also take place with the same probability $e^{-\\beta}$.  This jumping down can repeat with the quanta emitted from each belonging to the same mode.  However, once $A$ has changed significantly, the emission can be considered to have shifted to a different mode since $x$ will also have changed somewhat.  Similar remarks apply to jumps with $n\\Rightarrow n-2, n\\Rightarrow n-3,\\cdots$.\n\nIt is an assumption here that the chance of a jump is uninfluenced by whether the present level was the initial one or not, the chain of jumps being regarded as a quantum Markov process.   Obviously, with probability $1-e^{-\\beta}$ no jump takes place during the mode's time span, thus interrupting the chain.    Accordingly, the probability for $j$ successive jump-downs with given $x$ and $\\tilde m$ is\n\\begin{equation}\np_{\\downarrow j}= (1-e^{-\\beta})\\,e^{-\\beta j}.\n\\label{spontaneous} \n\\end{equation}\nThis probability distribution is already normalized: $\\sum_{j=0}^\\infty p_{\\downarrow j}=1$.\n\nThe probability $e^{-\\beta}$ refers to a spontaneous jump-down in the area spectrum with emission of one quantum.  There should also be an \\emph{elementary probability} for the hole in some state belonging to $l$  to jump-up to any state belonging to $u$~\\cite{BekSchiff} when a quantum of the appropriate $x$ and $\\tilde m$ is incident.  We denote this probability by $e^{-\\mu}$ with $\\mu$ also real and positive.  \n\nNow $e^{-\\mu}$ and $e^{-\\beta}$ are connected via the principle of detailed balance.  For example, we can consider the hole in equilibrium with a radiation bath at temperature $T_{\\rm H}$.   Then according to Planck the mean number of quanta incident on the black hole in a radiation mode labeled by $x$ is $(e^x-1)^{-1}$; we, of course, exclude the customary phase-space factor-it relates to the number of modes. The mean rate at which the hole jumps from the state in $l$ to that in $u$ must be proportional to this mean number.  It will be shown in Sec.~\\ref{sec:stimulated} that spontaneous emission must be accompanied by stimulated emission.  We assume that the hole's jump-down probability is uninfluenced by whether quanta in the relevant mode are already incident. Thus we can write\n\\begin{equation}\ne^{-\\mu}(e^x-1)^{-1}=e^{-\\beta}+e^{-\\beta}(e^x-1)^{-1},\n\\end{equation}\nwhich tells us that\n\\begin{equation}\ne^{-\\mu}=e^{-\\beta+x}.\n\\label{detailed}\n\\end{equation}\n\n\nWith $e^{-\\mu}$ in hand we can now calculate the probability of total absorption of $k$ quanta incident on the black hole in a single mode whose $x$ matches the spacing between levels $u$ and $l$.  Assuming that, as long as there are quanta that can be absorbed, the probability of a jump-up of the hole is independent of what happened before, we get for the probability of $k$ jump-ups in sequence with consequent absorption of $k$ quanta\n\\begin{equation}\np_{\\uparrow k}=(1-e^{-\\beta})e^{-\\mu k};\\qquad k=1,2,\\cdots .\n\\label{pup}\n\\end{equation}\nThe factor $(1-e^{-\\beta})$ again represents the probability that the hole does not experience a jump-down which would interrupt the sequence of $k$ jumps-up.   Note that in the present case the sum over $k$ in \\Eq{pup} need not be unity since different $k$s do not label different outcomes of one initial set up, but rather different initial set ups.\n\nOf course an incident quanta may fail to be absorbed, i.e. the black hole may not jump-up in its presence.  Now, the probability complementary to $e^{-\\mu}$ is $1-e^{-\\mu}$.  This is not the full contribution; the black hole may also spontaneously jump-down while failing to jump-up.  Thus the probability that the black hole avoids either and remains in its initial state despite the presence of an incident quantum is\n\\begin{equation}\np_0=(1-e^{-\\mu})(1-e^{-\\beta}).\n\\label{no}\n\\end{equation}\n\n\\section{Statistics of absorption and emission}\n\\label{sec:statistics}\n\nAs already hinted, \\Eq{spontaneous} can be regarded as the probability of \\emph{spontaneous} emission of $j$ quanta in the one mode considered.  We could write this as $p(j|0)$, where $p(m|n)$ is  generally defined as the \\emph{conditional probability} that $m$ quanta are emitted when $n$ are incident in the mode in question~\\cite{BekMeis}.  Of course we must require $\\sum_{m=0}^\\infty p(m|n)=1$ for all $n$.  Likewise we may interpret \\Eq{pup} to give $p(0|k)$, the probability that if $k$ quanta impinge on the black hole in a given mode, all are absorbed.  We now compute $p(m|n)$ for all other cases.\n\nSuppose $n$ indistinguishable quanta are incident on the black hole in one mode.  And suppose $k\\leq n$ of these fail to cause any jump-up of the black hole, and thus survive to occupy the outgoing counterpart of the incident mode. Assuming that each quantum acts independently and taking \\Eq{no} into account, the probability for this partial outcome must be $(1-e^{-\\mu})^k(1-e^{-\\beta})^k$.    The probability that the hole makes $n-k$ consecutive jump-ups as it absorbs the other $n-k$ incident quanta is evidently $e^{-\\mu (n-k)}$.   And the number of independent ways in which the $n$ incident quanta can be partitioned into absorbed and surviving quanta is $n!\/[(n-k)!k!]$.  \nWe thus get the \\emph{partial} probability (as always $0!=1$)\n\\begin{equation}\n\\frac{n!}{(n-k)!k!}(1-e^{-\\mu})^k(1-e^{-\\beta})^k e^{-\\mu (n-k)}.\n\\end{equation}\n\nThe number of quanta in the  outgoing counterpart of the said mode is $m$.  Of these $k\\leq m$ are surviving incident ones.  Thus the hole must emit $m-k$ quanta as it jumps down $m-k$ levels.  According to \\Eq{spontaneous} the  probability for this is $(1-e^{-\\beta})e^{-\\beta (m-k)}$.  The number of independent ways in which the $m!$ outgoers can be selected from surviving and freshly emitted quanta is $m!\/[(m-k)!k!]$.  We thus have the second partial probability\n\\begin{equation}\n\\frac{m!}{(m-k)!k!}(1-e^{-\\beta}) e^{-\\beta (m-k)}.\n\\end{equation}\nThe conditional emission probability $p(m|n)$ must obviously contain the product of the above partial probabilities.  In addition, the number $k$, being not observable, must be summed over from zero to the smaller of $n$ or $m$.  Thus\n\\begin{eqnarray}\np(m|n)&=&(1-e^{-\\beta})e^{-(\\mu n+\\beta m)}\\sum_{k=0}^{{\\rm min}(m,n)}\\frac{n!m!X^k}{(n-k)!(m-k)(k!)^2};\n\\label{pmn}\n\\\\\nX&\\equiv &(e^\\beta-1)(e^\\mu-1).\n\\label{X}\n\\end{eqnarray}\n\nTo check this result we first set $m=0$; we get $p(0,n)=(1-e^{-\\beta})e^{-\\mu n}$ which is what we would expect from \\Eq{pup}.  Setting rather $n=0$ we get $p(m,0)=(1-e^{-\\beta})e^{-\\beta m}$ which is the expected result for spontaneous emission, i.e., \\Eq{spontaneous}.  We next check that $p(m|n)$ is normalized.  Summing $p(m|n)$ over all $m$ and interchanging the two sums while respecting the constraints $k\\leq m,n$ gives\n\\begin{equation}\n\\sum_{m=0}^\\infty p(m|n)=(1-e^{-\\beta})e^{-\\mu n}\\sum_{k=0}^n\\frac{n!X^k}{(n-k)!(k!)^2}\\sum_{m=k}^\\infty \\frac{m!}{(m-k)!}e^{-\\beta m}.\n\\label{inter}\n\\end{equation}\nThe inner sum is done in Appendix A (\\Eq{first}); substituting it here gives\n\\begin{equation}\n\\sum_{m=0}^\\infty p(m|n)=\\sum_{k=0}^n\\frac{n!}{(n-k)!k!}(1-e^{-\\mu})^k e^{-\\mu (n-k)},\n\\end{equation}\nwhich equals unity by Newton's binomial theorem. We remark that the sum in \\Eq{pmn} is symmetric in $m$ and $n$ and can, in fact, be identified with the hypergeometric function ${}_2 F_1(-m,-n,1,X)$~\\cite{AbraSteg}.  The nonpositive integer nature of the first two arguments is responsible for cutting off the hypergeometric series and rendering it a polynomial in $X$; this turns out to be the Jacobi polynomial $P^{(0,-m-n-1)}(1-2X)$ (Eq.~15.4.6 of ref.~\\cite{AbraSteg}).  \n\nWe now express $e^{-\\beta}$ and $e^{-\\mu}$, the microscopic parameters of the black hole transition,  in terms of observable black hole parameters: $x$, as defined above, and $\\Gamma$, the so called absorptivity of the black hole in the said mode (also known as the grey-body factor or barrier tunneling coefficient).  First we calculate the mean number of quanta spontaneously emitted as a result of the jumps associated with the said mode (see \\Eq{spontaneous}):\n\\begin{equation}\n\\langle m\\rangle_{\\rm sp}=\\sum_{m=0}^\\infty (1-e^{-\\beta})\\,m\\,e^{-\\beta m}=(e^\\beta-1)^{-1}.\n\\label{mean0}\n\\end{equation} \nIn Hawking radiance this mean is $\\Gamma$ times what could be attributed to a \\emph{blackbody} with temperature $T=T_{\\rm H}$.  Although in the present paper the black hole spectrum is regarded as being made up of (broadened) lines, we can still use \\Eq{mean0} for the modes making up each line. \n\nNow an ideal black body would emit into the mode a Planck's mean number,\n\\begin{equation}\n\\langle m\\rangle_{\\rm bb}=(e^x-1)^{-1},\n\\end{equation}\nof quanta.  Thus the black hole spontaneously emits a mean number\n\\begin{equation}\n\\langle m\\rangle_{\\rm sp}=\\Gamma(e^x-1)^{-1}.\n\\label{spon}\n\\end{equation}\nComparing this with \\Eq{mean0} gives\n\\begin{equation}\ne^{-\\beta}=[1+(e^x-1)\\Gamma^{-1}]^{-1}.\n\\label{embeta1}\n\\end{equation}\n\nObviously for emission from a Schwarzschild black hole $x>0$ and $\\Gamma<1$; hence $e^{-\\beta}$ turns out to be smaller than unity, as it should.  For emission from the Kerr black hole $x=\\hbar(\\omega-\\tilde m\\Omega)\/T_{\\rm H}$.  For modes with $x>0$ the story is the same as for the Schwarzschild case.  Modes with $x<0$ are known as superradiant modes because incident radiation in such modes is amplified by the hole, so that $\\Gamma<0$ for every such mode.  For these modes it again follows from \\Eq{embeta1} that $e^{-\\beta}<1$.\n\nSubstitution of \\Eq{embeta1} and (\\ref{detailed}) into \\Eq{pmn} now gives us\n\\begin{equation}\np(m|n)=\\frac{(e^x-1)e^{xn}\\Gamma^{m+n}}{(e^x-1+\\Gamma)^{m+n+1}}\\ {}_2 F_1(-m,-n,1,X).\n\\label{pmn1}\n\\end{equation}\nThis expression for the conditional probability $p(m|n)$ is our central result here.  To recapitulate, it describes the statistics of black hole radiance in the particular mode characterized by the values of $x$ and $\\Gamma$.  It results from a set of simple assumptions as to the quantum jumps that the black hole can undergo, either spontaneous jumps, or those made possible by incident quanta in the same mode. \n\n\\section{Comparison with the early results}\n\\label{sec:comparison}\n\nThe consistency of that set of assumptions will now be verified by comparison of \\Eq{pmn1} with the expression for the conditional probability obtained by two independent methods that do not focus on the black hole's quantum structure. The $p(m|n)$ was first calculated in Ref.~\\cite{BekMeis} by applying the maximum entropy method of information theory to the statistics of radiation outgoing from a black hole which is bathed by external thermal radiation.  In Ref.~\\cite{PananWald} the same result was obtained by applying methods of quantum field theory in curved spacetime to the radiance.  Both of these approaches focus on the radiation, and treat the black hole as a fixed background.  It may be noted that an ideal material grey body also exhibits the same conditional probabilities~\\cite{BekSchiff}.\n\nThe results of Refs.~\\cite{BekMeis} and~\\cite{PananWald} can both be put in the form\n\\begin{eqnarray}\np(m|n)&=&\\frac{(e^x-1)e^{xn}\\Gamma^{m+n}}{(e^x-1+\\Gamma)^{m+n+1}}\\sum_{k=0}^{{\\rm min}(m,n)}\\frac{(m+n-k)!(X-1)^k}{(n-k)!(m-k)!k!};\n\\label{pmn2}\n\\\\\nX&=&\\frac{(e^x-1)^2e^{-x}(1-\\Gamma)}{\\Gamma^2},\n\\label{X2}\n\\end{eqnarray}\nwhere we have denoted the parameter also by $X$ since by \\Eqs{detailed} and (\\ref{embeta1}) it is equivalent to that in \\Eq{X}.\nNow the sum in \\Eq{pmn2} may be identified with \n\\begin{equation}\n\\frac{(m+n)!}{m!n!}{}_2F_1(-m,-n,-m-n,X-1);\n\\label{F}\n\\end{equation}\nthis is again a polynomial because the first two arguments are nonpositive integers.  The expression in \\Eq{F} is identical to the hypergeometric function appearing in \\Eq{pmn1}.\nHence, the expressions for $p(m|n)$ in \\Eqs{pmn1} and (\\ref{pmn2}) are identical.\n\nThis agreement shows that the assumptions made in Secs.~\\ref{sec:jump} and \\ref{sec:statistics} regarding the black hole area spectrum and transition dynamics are consistent with what is already known about black hole radiance, regardless of whether the hole is in vacuum or immersed in radiation.  In the previous section we composed $p(m|n)$ from the probability distributions for quanta which miss being absorbed in a black hole jump-up and for quanta which are emitted as a result of a hole jump-down.  In the next sections we shall rather compose $p(m|n)$ from the probability distributions for scattering, spontaneous and stimulated emission of quanta; this last approach was introduced in Ref.~\\cite{Fulfill}.\n\n\\section{Scattering}\n\\label{sec:scattering}\n\nImagine an incident quantum in a definite mode.  With probability $e^{-\\mu}$ it gets absorbed concurrently with a jump-up of the black hole.  This may be followed by a jump-down of the black hole or several such in sequence.  The total probability of these independent events is\n\\begin{equation}\n\\Gamma_0=e^{-\\mu}(1+e^{-\\beta}+e^{-2\\beta}+\\cdots)=e^{-\\mu}(1-e^{-\\beta})^{-1}.\n\\label{Gamma0}\n\\end{equation}\nEvidently, this is the probability that the incident quantum disappears into the hole with all possible consequences for the hole.  We denote it by $\\Gamma_0$ because in a real sense it is the true absorption probability for a quantum of the given mode whereas, as we shall show presently, $\\Gamma$ stands for a somewhat different thing.  \n\nAn external observer unaware of the inner workings of the black hole is surely aware of scattering off the hole's geometry.  What is the scattering probability distribution?  Following the above tack, we argue that since $\\Gamma_0$ is the absorption probability, $1-\\Gamma_0$ must be the probability that a quantum incident on the black hole in the given mode is scattered (reflected) into the outgoing form of the mode  regardless of what happened to the hole. Conservation of energy and angular momentum insure that the quantum remains associated with the same mode. Now if $n$ indistinguishable quanta are incident in one mode, the probability that a number $l\\leq n$ of these are scattered  is evidently\n\\begin{equation}\np_{\\rm sc}(l|n)=\\frac{n!}{l! (n-l)!}(1-\\Gamma_0)^l\\, \\Gamma_0{}^{n-l}.\n\\label{scatt}\n\\end{equation}\nUse of \\Eq{Gamma0} allows us to put the desired distribution in terms of observable parameters:\n\\begin{equation}\np_{\\rm sc}(l|n)=\\frac{n!}{l! (n-l)!} \\frac{e^{-\\mu n}e^{-\\beta l}(X-1)^{l} }{(1-e^{-\\beta})^n}.\n\\end{equation}\n\n\\section{Stimulated emission}\n\\label{sec:stimulated}\n\nIf we replace $e^{-\\mu}$ in \\Eq{Gamma0} by means of \\Eq{detailed} and then replace $e^{-\\beta}$ in the result with help of \\Eq{embeta1} we get\n\\begin{equation}\n\\Gamma=\\Gamma_0(1-e^{-x})\n\\label{Gamma1}.\n\\end{equation}\nThis result already shows that a black hole is capable of stimulated emission~\\cite{BekMeis,Fulfill}.  To see this consider how $\\Gamma$ is calculated in practice.  One imagines a classical wave in the form of a mode function of the appropriate field directed onto the hole; $1-\\Gamma$ is identified with the fraction of the energy of the initial wave which is returned outward as a result of the scattering by the hole.   Recall that $\\Gamma_0$, being a probability, is always positive.  Now for a mode with $x<0$, \\Eq{Gamma1} predicts that $\\Gamma<0$ so that $1-\\Gamma>1$: the incident wave is thus predicted to be amplified, a sure sign of stimulated emission.  The corresponding modes are called super radiant, and are well studied numerically.\n\nTurn now to a mode with $x>0$. \\Eq{Gamma1} shows that $0<\\Gamma<1$ so that $1-\\Gamma$ \\emph{exceeds} the scattering probability $1-\\Gamma_0$ (the ratio of the two last two being independent of the wave's initial amplitude).  How is this possible?  Only if the black hole is stimulated by the incident wave to emit the same kind of wave and thus strengthen the outgoing wave in direct proportion to the incident wave's strength.  In this non-superradiant mode  the outgoing wave is weaker than the incident one, but not as weak as would be expected from the size of $\\Gamma_0$.  We must conclude~\\cite{BekMeis,Fulfill} that stimulated emission occurs in these modes also, but with strength insufficient to actually amplify the incident wave.  \n\nTurning to our main concern, we ask what is $p_{\\rm st}(m|n)$, the conditional probability for emission of $m$ quanta given that $n$ are incident in the same mode?  In ref.~\\cite{Fulfill} it was obtained by judiciously decomposing $p(m|n)$ into a the scattering contribution from \\Eq{scatt} and a part which could be interpreted in terms of emission.  Here we shall derive  $p_{\\rm st}(m|n)$ using an approach like the one we followed to obtain $p(m|n)$ in Sec.~\\ref{sec:statistics} of the present paper.\n\nStimulated emission means that each of the $n$ incident quanta may be responsible for emission of some of the outgoing $m$ quanta.  In how many ways can one associate the outgoing quanta, each to some incident one, all this in harmony with the bosonic character of the quanta?  Recall the standard textbook question, in how many ways can one arrange $m$ bosons in $g$ cells given that the bosons are identical?~\\cite{LLSP}.  The answer is \n\\begin{equation}\n\\frac{(m+g-1)!}{m!(g-1)!}.\n\\label{comb}\n\\end{equation}  \nThis last problem maps directly into ours; we get the desired number by replacing $g\\Rightarrow n$ above.\n\nWe shall assume that the dynamics of black-hole jump-down and quantum emission is the same whether occurring spontaneously or whether it is induced by an incident quantum.  This is in keeping with what we know from atomic physics (equality of the properly defined Einstein coefficients for spontaneous and stimulated emission).  The one difference here is that it is possible for the black hole to be induced to make several jump-downs in sequence and place the consequently emitted quanta in the same mode.  (This last is very rare, if at all possible, in atomic physics.)  To the $m_j$ emitted quanta associated with incident quantum $j$ we must associate a probability $e^{-\\beta m_j}$ corresponding to $m_j$ successive black hole jump-downs.  After those the jumping-down is arrested with probability $1-e^{-\\beta}$.  Thus the overall probability $p_{\\rm st}(m|n)$ is given by\n\\begin{equation}\n(1-e^{-\\beta})e^{-\\beta m_1}\\cdot (1-e^{-\\beta})e^{-\\beta m_2} \\cdots (1-e^{-\\beta})e^{-\\beta m_{n}}.\n\\end{equation}\nIn light of \\Eq{comb} and the fact that $\\sum_j m_j=m$, we have our final formula\n\\begin{equation}\np_{\\rm st}(m|n)=\\frac{(m+n-1)!}{m!(n-1)!}(1-e^{-\\beta})^n e^{-\\beta m}.\n\\label{pst}\n\\end{equation}\nOf course this formula can be used only for $n\\geq 1$.\n\nTo analyze this result we first compute the \\emph{conditional mean} number of quanta emitted given that $n$ are incident:\n\\begin{equation}\n\\langle m|n\\rangle_{\\rm st}=\\frac{(1-e^{-\\beta})^{n}}{(n-1)!} \\sum_{m=0}^\\infty \\frac{(m+n)!}{m!}\\, e^{-\\beta (m+1)}\n\\label{sum3}\n\\end{equation}\nUsing \\Eq{sum3'} of the Appendix we have\n\\begin{equation}\n\\langle m|n\\rangle_{\\rm st}=\\frac{n}{e^\\beta-1}=n\\langle m\\rangle_{\\rm sp}.\n\\end{equation}\nThis is entirely analogous to the result in atomic physics that the mean number of quanta from stimulated emission is proportional to the number of incident quanta, with the proportionality coefficient equal to the mean number from spontaneous emission.  Our result here thus strengthens the claim that a black hole is capable of stimulated emission even for non-superradiant modes.\n\nActually in atomic physics one computes the \\emph{probability} of stimulated emission of \\emph{one} photon given that $n$ are incident, and finds it to be $n$ times the probability of one-photon spontaneous emission.   Were we to do likewise here we would get\n\\begin{equation}\n\\frac{p_{\\rm st}(1|n)}{p(1|0)}=n(1-e^{-\\beta})^{n-1}.\n\\end{equation}\nThis corresponds to the atomic physics result only for $n=1$ or when $\\beta\\to\\infty$ for any $n\\geq 2$.  Now by \\Eq{spon} $\\beta\\to\\infty$ is equivalent to $x\\to\\infty$, i.e., to the case $\\hbar\\omega\\gg T$ (Wien regime).  In the intermediate and the Rayleigh-Jeans regimes the coefficient is smaller than that in the atomic physics calculation.  We interpret this discrepancy to reflect the fact that the mentioned atomic physics result is a first-order perturbation one, whereas \\Eq{pst} here is nonperturbative.  This last is obvious since the formula in \\Eq{spon} works for multi-quanta emission which would require application of arbitrarily high order perturbation theory.\n\n\\section{Composition of emission with scattering}\n\\label{sec:composition}\n\nLet us now compose the distribution $p_{\\rm st}(m|n)$ with that for spontaneous emission, $p(m|0)$, to get the conditional probability distribution for generic emission, $p_{\\rm em}(m|n)$:\n\\begin{equation}\np_{\\rm em}(m|n)=\\sum_{k=0}^m p_{\\rm st}(k|n)\\, p(m-k|0),\n\\end{equation}\nwhich equals\n\\begin{equation}\n(1-e^{-\\beta})^{n+1}e^{-\\beta m}\\sum_{k=0}^m \\frac{(n+k-1)!}{k!(n-1)!}.\n\\label{sum4}\n\\end{equation}\nThe sum is worked out in the Appendix (\\Eq{second}); we find\n\\begin{equation}\np_{\\rm em}(m|n)=\\frac{(m+n)!}{m!n!}(1-e^{-\\beta})^{n+1} e^{-\\beta m}.\n\\end{equation}\n\nWe note that $p_{\\rm em}(m|n)=p_{\\rm st}(m|n+1)$.  This accords with what happens in atomic physics: the total emission in the presence of $n$ quanta is as if $n+1$ quanta were responsible for purely stimulated emission, the extra unity being ascribed to the effect of vacuum fluctuations. \n\nWe now compose the distributions $p_{\\rm sc}$ and $p_{\\rm em}$; this should give us $p(m|n)$:\n\\begin{equation}\np(m|n)=\\sum_{k=0}^{{\\rm min}(m,n)}  p_{\\rm em}(m-k|n)\\,p_{\\rm sc}(k|n),\n\\end{equation}\nwhere the upper limit of the summation reflects the fact that no more quanta can be scattered than the number incident but also cannot exceed the total number of outgoing quanta.  Thus\n\\begin{equation}\np(m|n)=(1-e^{-\\beta})e^{-(\\mu n+\\beta m)}\\sum_{k=0}^{{\\rm min}(m,n)}  \\frac{(m+n-k)!(X-1)^{k}}{k!(m-k)! (n-k)!}  \n\\label{pmn3}\n\\end{equation}\nThe coefficient of the sum here is exactly equivalent to that in \\Eq{pmn2}.  Thus by focusing on the scattering contribution we have here derived an alternative form of $p(m|n)$, identical to that found in refs.~\\cite{BekMeis,PananWald}, and which is also known to be equivalent to the $p(m|n)$ deduced in Sec.~\\ref{sec:statistics} of this paper (see Sec.~\\ref{sec:comparison}). \n\n\\section{Summary}\n\nIt is widely accepted that a thermally radiating system cannot disclose details about its internal dynamics due to the loss of coherence implicit in the thermalization.  A black hole may be different in that, unlike ordinary matter, it is not demonstrably a collection of independent systems.   In this paper we have leaned on the hypothesis that a Kerr black hole's horizon area has a discrete spectrum to derive the black hole's response to incident radiation, specifically the conditional probability distribution for the number of emitted quanta.  We reproduce in two different ways the known probability distribution, obtaining to boot a closed form of it in terms of an hypergeometric function.  This accomplishment of the ``atomic'' model of the black hole shows that the widely hypothesized discreteness of the area spectrum is consistent with the expected statistics of the black hole radiance.   \n\nOur approach here is related to the approach to quantum gravity of a black hole based on its quasinormal modes~\\cite{BekMG,Hod,Maggiore}.   Quasinormal modes, however, are purely classical responses of the black hole to perturbations, and one cannot immediately discount the possibility that they are irrelevant to questions of quantum structure.  Nevertheless, the quasinormal mode approach does point to a discrete horizon area spectrum, often a uniformly spaced one~\\cite{Skakala},~ and so agrees with our conclusion here that the black hole response to external radiation is consistent with a discrete area spectrum.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}