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+{"text":"\\section{Properties of the operators $\\{\\Pi_{\\mu}\\}$} \nHere we show that the operators in Eq.~\\eqref{eq:pilambda} are mutually orthogonal projectors that add up to identity. To this end we use the \\emph{grand orthogonality theorem} (a consequence of Schur's lemma) of representation theory.\n\\begin{theorem}[Grand orthogonality theorem] (See \\cite[Sec. 2.2.]{serre1977linear})\n    Given irreducible representations $\\rho_\\mu$ and $\\rho_\\nu$, we have that: \n    \\begin{align}\n        \\sum_{g \\in S_n} \\rho_\\mu(g)^\\dag_{ij} \\rho_\\nu(g)_{k\\ell} &= \\frac{n!}{d_\\mu} \\delta_{ik} \\delta_{j \\ell} \\delta_{\\mu \\nu}. \n    \\end{align} \n\\label{thm:got}\n\\end{theorem}\nUsing Theorem \\ref{thm:got} we obtain the following identity for any fixed $h \\in S_n$.\n\\begin{align*}\n    \\sum_{g \\in S_n} \\chi_\\nu(g) \\chi_\\mu(g^{-1} h) &= \\sum_{g \\in S_n} \\sum_{i,j} \\rho_\\nu(g)_{ii} \\rho_\\mu(g^{-1}h)_{jj} \\\\ &= \\sum_{g \\in S_n} \\sum_{ijk}\\rho_\\nu(g)_{ii} \\rho_\\mu(g^{-1})_{jk} \\rho_\\mu(h)_{kj} \\\\ &=  \\sum_{ijk}  \\left( \\sum_{g \\in S_n} \\rho_\\nu(g)_{ii} \\rho_\\mu(g)^\\dag_{jk} \\right) \\rho_\\mu(h)_{kj} \\\\ \n    &= \\delta_{\\mu \\nu} \\sum_{ijk} \\delta_{ij} \\delta_{ik} \\frac{n!}{d_\\nu} \\rho_\\mu(h)_{kj} = \\delta_{\\mu \\nu} \\sum_i \\frac{n!}{d_\\nu} \\rho_\\mu(h)_{ii} = \\delta_{\\mu \\nu} \\frac{n!}{d_\\nu} \\chi_\\mu(h).\n    \\end{align*}\nIt follows that $\\{\\Pi_{\\mu}\\}$ are mutually orthogonal projectors. \n\\begin{align}\n    \\Pi_\\nu \\Pi_\\mu &= \\frac{d_\\mu d_\\nu}{(n!)^2} \\sum_{gh} \\chi_\\nu(g) \\chi_\\mu(h) \\rho(gh) = \\frac{d_\\mu d_\\nu}{(n!)^2} \\sum_{g \\ell} \\chi_\\nu(g) \\chi_\\mu(g^{-1} \\ell) \\rho(\\ell) = \\delta_{\\mu \\nu} \\frac{d_\\mu}{n!} \\sum_\\ell \\chi_\\mu(\\ell) \\rho(\\ell) = \\delta_{\\mu \\nu} \\Pi_\\mu.\n\\end{align}\nUsing the fact that $d_\\mu = \\chi_\\mu(e)$, where $e \\in S_n$ is the identity, we obtain\n\\begin{align}\n    \\sum_{\\mu \\vdash n} \\Pi_\\mu &= \\sum_{g \\in S_n} \\frac{1}{n!} \\rho(g) \\sum_{\\mu \\vdash n} d_\\mu \\chi_\\mu(g) = \\sum_{g \\in S_n} \\frac{1}{n!} \\rho(g) \\sum_{\\mu \\vdash n} \\chi_\\mu(e) \\chi_\\mu(g).\n\\label{eq:sumpi}\n\\end{align}\n\nThe column-orthogonality of characters (stated below) implies that\n\\begin{align}\n    \\sum_\\mu \\chi_\\mu(e) \\chi_\\mu(g) &= n! \\delta_{eg},\n\\end{align}\nand substituting this into Eq.~\\eqref{eq:sumpi} gives $\\sum_\\mu \\Pi_\\mu = I $.\n\n\n\n\\begin{lemma}[Column-orthogonality of characters] (See \\cite[Thm 16.4]{alperin2012groups})\nFor any $h,g\\in S_n$ we have\n    \\begin{align}\n    \\sum_{\\mu \\vdash n} \\chi_\\mu(h) \\chi_\\mu(g) &= \\begin{cases}\n    |Z(g)|  &\\text{ if $h$ and $g$ are conjugate.}\\\\ \n    0 & \\text{ otherwise. }\n    \\end{cases}\n\\end{align}\nwhere $Z(g)$ is the centralizer of $g$ in $S_n$ (the set of all group elements that commute with $g$).\n\\end{lemma}\n\n\n\n\\section{Beals' quantum Fourier transform over the symmetric group}\nFor completeness, here we review Beals' efficient algorithm for the quantum Fourier transform over the symmetric group \\cite{beals1997quantum} (see also \\cite{Moore03} for generalizations to other nonabelian groups).\n\n  Let $n\\geq 2$ be an integer and consider the symmetric group $S_n$ of permutations on $n$ objects. Also let $T=\\{\\tau_1,\\tau_2,\\ldots, \\tau_n\\}$ be a transversal of the left cosets of $S_{n-1}$ in $S_n$. Here $S_{n-1}$ is regarded as the subgroup of $S_n$ that does not permute the $n$-th object and transversality means that $S_n$ is a union of cosets $\\tau_1S_{n-1},\\ldots,\\tau_n S_{n-1}$.\n\nIt will be convenient to fix some notation and conventions concerning the function whose Fourier transform we are interested in. We shall fix a unitary matrix representation of each irreducible representation of $S_n$. In particular, we choose the so-called Young Orthogonal representation (more on this below) and write $\\rho_{\\lambda}(\\pi)$ for the matrix corresponding to the irreducible representation labeled by $\\lambda \\vdash n$ evaluated at $\\pi\\in S_n$, and $d_{\\lambda}$ for its dimension. \n\nFor a function $f:S_n \\rightarrow \\mathbb{C}$ we have\n \\begin{equation}\n \\mathrm{QFT}_n\\sum_{\\sigma\\in S_n} f(\\sigma)|\\sigma\\rangle=\\sum_{\\omega \\vdash n} \\sum_{i,j \\in d_\\omega} (\\hat{f}(\\omega))_{ij}  |\\omega, i,j\\rangle, \n\\label{eq:ft11}\n\\end{equation}\nwhere\n\\begin{equation}\n \\hat{f}(\\omega)=\\sqrt{\\frac{d_{\\omega}}{n!}} \\sum_{\\sigma\\in S_n} f(\\sigma)\\rho_\\omega(\\sigma).\n\\label{eq:ft22}\n \\end{equation}\nFor ease of notation in this section we include the normalization factor $\\sqrt{\\frac{d_{\\omega}}{n!}}$ in Eq.~(\\ref{eq:ft22}) rather than in Eq.~\\eqref{eq:ft11} which differs from our convention in the main text (Cf. Eqs.~(\\ref{eq:ft1},\\ref{eq:ft2})). However the quantum fourier transform $\\mathrm{QFT}_n$ is defined in exactly the same way.\n\nNote that the basis vectors in the Fourier basis can be labeled by triples from the set $\\Omega=\\{(\\omega, i,j): \\omega\\vdash n, 1\\leq i,j\\leq d_{\\omega}\\}$, since $|\\Omega|=\\sum_{\\omega\\vdash n} d_{\\omega}^2 =|S_n|$.\nSince the left cosets of $S_{n-1}$ partition $S_n$, we may write\n\\begin{equation}\nf(g)=\\sum_{j=1}^{n} F^{j}(g) \\qquad \\qquad F^{j}(g)\\equiv \\begin{cases} f(g) &, \\text{if }  g\\in \\tau_j S_{n-1}\\\\ 0 &, \\text{otherwise} \\end{cases}.\n\\label{eq:defF}\n\\end{equation}\nAlso define $f_j:S_{n-1}\\rightarrow \\mathbb{C}$ for $1\\leq j\\leq n$ by \n\\begin{equation}\nf_j(h)\\equiv f(\\tau_j h) \\qquad \\quad h\\in S_{n-1}.\n\\label{eq:deff}\n\\end{equation}\n\nYoung's orthogonal representation has a certain \\textit{adapted} property that allows us to express Eq.~\\eqref{eq:ft22} in terms of Fourier transforms of the functions $f_j$ for $1\\leq j\\leq n$ in a simple way. In particular, \n\\begin{equation}\n\\hat{f}(\\lambda)=\\sum_{k=1}^{n} \\rho_{\\lambda}(\\tau_k) \\bigoplus_{\\lambda^{-}\\in \\Phi(\\lambda) } \\sqrt{\\frac{d_{\\lambda}}{n\\cdot d_{\\lambda^{-}}}}\\cdot \\hat{f_k} (\\lambda^{-}),\n\\label{eq:dirsum}\n\\end{equation}\nwhere \n\\[\n\\Phi(\\lambda)=\\{\\lambda^{-}\\vdash (n-1) : \\lambda^{-}\\leq \\lambda\\}\n\\]\nis the set of partitions of $n-1$ whose diagram differs from $\\lambda$ in exactly one box. Note that the Fourier transforms $\\hat{f_k}$ appearing on the right hand side of Eq.~\\eqref{eq:dirsum} are over the group $S_{n-1}$.\n\n\n\nTo describe Beals' QFT algorithm it will be convenient to first define some unitary transformations that are used in the construction. In the following we shall assume that we have enough qubits to encode certain classical strings as computational basis states.\n\n\n Let $W$ denote the unitary which implements a classical reversible circuit that, given a  permutation $\\pi\\in S_n$, computes its factorization:\n\\[\nW|\\pi\\rangle=|\\tau_j\\rangle|h\\rangle,\n\\]\nwhere $j\\in [n]$ and $h\\in S_{n-1}$\nsatisfy $\\pi = \\tau_j h$. Note that this factorization is unique.\n\nFor $\\pi \\in S_n$ let $M(\\pi)$ be a unitary such that\n\\[\nM(\\pi) \\sum_{\\lambda\\vdash n}\\sum_{p,q=1}^{d_{\\lambda}} (A_{\\lambda})_{p,q}|\\lambda,p,q\\rangle= \\sum_{\\lambda\\vdash n} \\sum_{p,q=1}^{d_{\\lambda} }(\\rho_{\\lambda}(\\pi)A_{\\lambda})_{p,q}|\\lambda,p,q\\rangle\n\\]\nHere $(A_{\\lambda})_{p,q}$ are complex coefficients. Note that this can be implemented via a controlled-$\\rho_{\\lambda}(\\pi)$ operation on the $p$ register (where the control is $\\lambda$).\n The unitary $M(\\pi)$ acts trivially outside the subspace spanned by basis vectors $\\{|\\lambda,p,q\\rangle: \\lambda \\vdash n, p,q\\in [d_{\\lambda}]\\}$\n\nFor each irreducible representation $\\sigma$ of $S_{n-1}$ there is a set of irreps $\\lambda$ of $S_n$ such that the restriction of $\\rho_{\\lambda}$ to $S_{n-1}$ contains $\\sigma$. These correspond to the partitions $\\lambda$ that differ from $\\sigma$ by adding one box. Moreover, each matrix element $p,q$ of $\\rho_{\\sigma}$  (where $p,q\\leq d_{\\sigma}$) is identified with a unique matrix element $p',q'$ of $\\rho_{\\lambda}$ (Here $p',q'$ depend on $p,q$ as well as $\\lambda$, though our notation suppresses these dependencies).\n\nLet $U$ be a unitary such that for any $\\sigma \\vdash (n-1)$ and $p,q\\leq d_{\\sigma}$ we have\n\\begin{equation}\nU|0\\rangle|\\sigma,p,q\\rangle=|0\\rangle\\sum_{\\lambda \\vdash n: \\sigma \\leq \\lambda} \\sqrt{\\frac{d_{\\lambda}}{n\\cdot d_{\\sigma}}}|\\lambda,p',q'\\rangle \n\\label{eq:U}\n\\end{equation}\nand such that \n\\begin{equation}\nU|\\tau_i\\rangle|\\sigma,p,q\\rangle=|\\tau_i\\rangle|\\sigma,p,q\\rangle  \\quad \\qquad i\\in [n].\n\\label{eq:pkinvariant}\n\\end{equation}\n It is possible to construct such a unitary because the states appearing on the RHS of Eq.~\\eqref{eq:U} for different values of $\\sigma,p,q$ are orthonormal. However the action of $U$ in the rest of the Hilbert space is constrained by unitarity. The following lemma describes a property of $U$ that follows from this constraint.\n\\begin{lemma}\nLet $P$ project onto the subspace spanned by basis states than encode partitions of $n-1$, that is, \n\\[\n\\mathrm{span}\\{|w\\rangle|\\sigma,p,q\\rangle: w\\in \\{0,\\tau_1,\\ldots, \\tau_n\\}, \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}.\n\\]\nSuppose $g:S_n\\rightarrow \\mathbb{C}$ is any function such that $g(\\pi)=0$ for all $\\pi\\in S_{n-1}$ (regarded as the subgroup of $S_n$ that fixes $n$). Then $PU^{\\dagger}|0\\rangle|\\hat{g}\\rangle=0$.\n\\label{lem:orthog}\n\\end{lemma}\n\\begin{proof}\nBelow we write $P=\\sum_{k=0}^{n}P_k$ where $P_0=\\mathrm{span}\\{|0\\rangle|\\sigma,p,q\\rangle: \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}$ and $P_k=\\mathrm{span}\\{|\\tau_k\\rangle|\\sigma,p,q\\rangle: \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}$ for $k\\in [n]$.\n\nFirst suppose that $\\hat{r}$ is the Fourier transform of some (arbitrary) function $r:S_{n-1}\\rightarrow \\mathbb{C}$. Define $R:S_n\\rightarrow \\mathbb{C}$ such that $R(\\pi)=r(\\pi)$ for all $\\pi\\in S_{n-1}$, and $R(\\pi)=0$ otherwise. Then from Eq.~\\eqref{eq:U} we have\n\\begin{equation}\n|0\\rangle\\sum_{\\sigma\\vdash n-1,p,q} (\\hat{r}(\\sigma))_{pq}|\\sigma,p,q\\rangle=U^{\\dagger}|0\\rangle\\sum_{\\lambda\\vdash n, p',q'} (\\hat{R}(\\lambda))_{p',q'}|\\lambda,p',q'\\rangle. \n\\label{eq:rR}\n\\end{equation}\nNow consider a function $g:S_n\\rightarrow \\mathbb{C}$ as in the statement of the Lemma. Since \n$g(\\pi)=0$ for all $\\pi\\in S_{n-1}$ we have $\\langle g|R\\rangle=0$ and therefore $\\langle \\hat{g}|\\hat{R}\\rangle=0$ and therefore \n$U^{\\dagger}|0\\rangle|\\hat{g}\\rangle$ is orthogonal to any state of the form on the LHS of Eq.~\\eqref{eq:rR}. Noting that the LHS of Eq.~\\eqref{eq:rR} is an arbitrary state in the image of $P_0$, we see that \n\\begin{equation}\nP_0U^{\\dagger}|0\\rangle|\\hat{g}\\rangle=0.\n\\label{eq:p0}\n\\end{equation}\n Then \n\\begin{align*}\nPU^{\\dagger}|0\\rangle|\\hat{g}\\rangle&=(\\sum_{k=1}^{n}P_k)U^{\\dagger}|0\\rangle|\\hat{g}\\rangle+P_0U^{\\dagger}|0\\rangle|\\hat{g}\\rangle\\\\\n&=(\\sum_{k=1}^{n}P_k)U^{\\dagger}|0\\rangle|\\hat{g}\\rangle\\\\&=(\\sum_{k=1}^{n}P_k)|0\\rangle|\\hat{g}\\rangle\\\\\n&=0\\\\\n\\end{align*}\nwhere in the first step we used Eq.~\\eqref{eq:p0} and in the following step we used Eq.~\\eqref{eq:pkinvariant}. \n\\end{proof}\n\nOur definitions of $U, f_k, F^k$ and $M(\\tau_k)$ are chosen so that the following holds.\n\\begin{claim}\nFor $1\\leq k\\leq n$ we have $M(\\tau_k)U|0\\rangle|\\hat{f_k}\\rangle=|0\\rangle|\\hat{F^k}\\rangle$.\n\\label{claim:Fkfk}\n\\end{claim}\n\\begin{proof}\nFollows by combining Eqs.~(\\ref{eq:dirsum}, \\ref{eq:U}, \\ref{eq:defF}, \\ref{eq:deff}).\n\\end{proof}\nFinally, for $k\\in [n]$ let $V_k$ be the unitary that acts as\n\\begin{align}\nV_k|0\\rangle|\\sigma,p,q\\rangle &=|\\tau_k\\rangle|\\sigma,p,q\\rangle \\\\\nV_k|\\tau_k\\rangle|\\sigma,p,q\\rangle &=|0\\rangle|\\sigma,p,q\\rangle,\n\\end{align}\nfor all $\\sigma\\vdash n-1$ and $1\\leq p,q\\leq d_{\\sigma}$,\nand which acts as the identity on all other computational basis states.\n\nThe quantum Fourier transform over the symmetric group is described in Algorithm \\ref{alg:qft}.\n\\begin{algorithm}[H]\n\\caption{Implements a unitary $\\mathrm{QFT_n}$\\label{alg:qft}}\n\\hspace*{\\algorithmicindent}  \\hspace{-22pt}\\textbf{Input:}  A state $|f\\rangle=\\sum_{g\\in S_n} f(g)|g\\rangle$.\\\\\n\\hspace*{\\algorithmicindent} \\hspace{-27pt} \\textbf{Output:} The state $|\\hat{f}\\rangle=\\mathrm{QFT}_n|f\\rangle$ corresponding to the Fourier transform of $f$ over $S_n$.\\\\\n\n\\begin{algorithmic}[1]\n\t\\State{$|\\phi\\rangle\\leftarrow W|\\phi\\rangle$} \\Comment{$|\\phi\\rangle=\\sum_{j=1}^{n} |\\tau_j\\rangle\\sum_{h\\in S_{n-1}} f_j(h)|h\\rangle$}.\n\t\\State{$|\\phi\\rangle \\leftarrow (I\\otimes \\mathrm{QFT}_{n-1})|\\phi\\rangle$} \\Comment{$|\\phi\\rangle=\\sum_{j=1}^{n} |\\tau_j\\rangle \\sum_{\\sigma \\vdash n-1} (\\hat{f_j}(\\sigma))_{pq} |\\sigma,p,q\\rangle$}\n\t\t\t\\For{$k=1$ to $n$}\n\t\t\\State{$|\\phi\\rangle\\leftarrow M(\\tau_k)^{-1}|\\phi\\rangle$ \n\t\t}\n\t\t\t\\State{$|\\phi\\rangle\\leftarrow UV_kU^{\\dagger}|\\phi\\rangle$}\n\t\t\\State{$|\\phi\\rangle\\leftarrow M(\\tau_k)|\\phi\\rangle$ } \n\t\t\t      \\EndFor \\Comment{$|\\phi\\rangle=|0\\rangle|\\hat{f}\\rangle$ (see Lemma \\ref{lem:forloop})} \n\\State{\\textbf{return} the second register $|\\hat{f}\\rangle$}.\n\t\t\\end{algorithmic}\n\\end{algorithm}\n\nThe following lemma shows that the algorithm performs the quantum Fourier transform, as claimed.\n\n\\begin{lemma}\nFor $0\\leq k\\leq n$, the state after the $k$th iteration of the for loop in Algorithm \\ref{alg:qft} is\n\\begin{equation}\n|\\phi_k\\rangle=|0\\rangle\\sum_{j=1}^{k}|\\hat{F^{j}}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\label{eq:phik}\n\\end{equation}\nWhen $k=0$ this describes the state before the first iteration (in this case the first term is not present). For $k=n$ we have $|\\phi_n\\rangle=|0\\rangle\\sum_{j=1}^{n}|\\hat{F^{j}}\\rangle=|0\\rangle|\\hat{f}\\rangle$.\n\\label{lem:forloop}\n\\end{lemma}\n\\begin{proof}\nBy induction on $k$. The base case $k=0$ corresponds to the initial state $\\sum_{j=1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle$. \n\n Now suppose the state after the $(k-1)$-th iteration is given by $|\\phi_{k-1}\\rangle$ as described by Eq.~\\eqref{eq:phik}. Then after line 4 of Algorithm \\ref{alg:qft} the state is\n\\begin{align}\nM(\\tau_k^{-1})|\\phi_{k-1}\\rangle&=|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^{j}}\\rangle+\\sum_{j=k}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\end{align}\nFor each $j\\leq k-1$, the state $M(\\tau_k^{-1})|\\hat{F^j}\\rangle$ is the Fourier transform of a function \n$g(\\pi)=F^j(\\tau_k \\pi)$ which is zero on $S_{n-1}$.\nSo for $j\\leq k-1$, by Lemma \\ref{lem:orthog} we have $P U^{\\dagger}|0\\rangle M(\\tau_k^{-1})|\\hat{F^j}\\rangle =0$ where $P$ projects onto basis states that encode partitions of $n-1$. Therefore $V_k$ acts trivially on $U^{\\dagger}|0\\rangle M(\\tau_k^{-1})|\\hat{F^j}\\rangle$  for all $j\\leq k-1$, i.e., \n\\begin{align}\nV_kU^{\\dagger}M(\\tau_k^{-1})|\\phi_{k-1}\\rangle&=U^{\\dagger}|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^j}\\rangle+\\sum_{j=k}^{n} V_k|\\tau_j\\rangle |\\hat{f_j}\\rangle\\\\\n&=U^{\\dagger}|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^j}\\rangle+|0\\rangle|\\hat{f_k}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\end{align}\nand, applying $M(\\tau_k)U$ to the above gives\n\\begin{align}\n|\\phi_k\\rangle&=|0\\rangle\\sum_{j=1}^{k-1}|\\hat{F^j}\\rangle+M(\\tau_k)U|0\\rangle|\\hat{f_k}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle\\\\\n&=|0\\rangle\\sum_{j=1}^{k}|\\hat{F^j}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle\n\\end{align}\nwhere we used Claim \\ref{claim:Fkfk}. This completes the induction, and the proof.\n\\end{proof}\n\n\n\\section{ Generalized phase estimation}\nIn this section we will show that the circuits from Figure~\\ref{fig:gpe} implement weak Fourier sampling and generalized phase estimation respectively. The following Lemma shows that the circuit in Figure \\ref{fig:gpe}(a) implements weak Fourier sampling. We will make use of the inverse Fourier transform $\\mathrm{QFT}_n^{\\dagger}$ which acts as:\n\\begin{equation}\n\\mathrm{QFT}_n^{\\dagger} \\sum_{\\omega \\vdash n}\\sum_{1\\leq i,j\\leq d_{\\omega}} (c(\\omega))_{ij} |\\omega, i,j\\rangle=\\sum_{\\sigma\\in S_n}\\sum_{\\lambda \\vdash n} \\sqrt{\\frac{d_\\lambda}{n!}} \\trace{\\left(c(\\lambda) \\rho_{\\lambda}(\\sigma)^{\\dagger}\\right)}|\\sigma\\rangle.\n\\label{eq:invf}\n\\end{equation}\n\n \\begin{lemma}\n The POVM $M_L$ can be implemented by applying $\\mathrm{QFT}_n$, then performing a projective measurement of the representation label $\\omega$, then appplying $\\mathrm{QFT}_n^{\\dagger}$.\n \\label{Lemma:partition_measurement}\n \\end{lemma}\n \\begin{proof}\n Let a partition $\\lambda \\vdash n$ be given.  Let $P_{\\lambda}$ denote the projector such that $P_{\\lambda}|\\omega,i,j\\rangle=\\delta_{\\omega, \\lambda} |\\omega,i,j\\rangle.$ It suffices to show that for any state $|\\psi\\rangle\\in \\mathcal{H}$ we have $\\Pi^L_{\\lambda}|\\psi\\rangle=\\mathrm{QFT}_n^{\\dagger} P_{\\lambda} \\mathrm{QFT}_n|\\psi\\rangle$. To this end let $|\\psi\\rangle=\\sum_{\\sigma \\in S_n} f(\\sigma)|\\sigma\\rangle$. Using Eq.~\\eqref{eq:pilambda} we get\n \\begin{align}\n \\Pi_{\\lambda}|\\psi\\rangle& =\\frac{d_{\\lambda}}{n!} \\sum_{\\sigma\\in S_n}\\sum_{\\alpha\\in S_n}\\chi_{\\lambda}(\\alpha)f(\\sigma)|\\alpha \\sigma\\rangle\\\\\n &=\\frac{d_{\\lambda}}{n!}\\sum_{\\sigma,\\beta\\in S_n} \\chi_{\\lambda}(\\beta\\sigma^{-1})f(\\sigma)|\\beta\\rangle =\\frac{d_{\\lambda}}{n!}\\sum_{\\sigma,\\beta\\in S_n} \\chi_{\\lambda}(\\sigma \\beta^{-1})f(\\sigma)|\\beta\\rangle,\n \\label{eq:chif1}\n \\end{align}\n where in the last line we used the fact that $\\chi_\\lambda (g)=\\chi_{\\lambda}(g^{-1})$ for all $g\\in S_n$  (this can be seen for example using the fact that group characters are class functions and $g$ and $g^{-1}$ are always in the same conjugacy class since their cycle structure coincides). Now using Eq.~\\eqref{eq:chif1} and the fact that \n \\[\n \\chi_\\lambda(\\sigma \\beta^{-1})=\\trace{\\left(\\rho_\\lambda(\\sigma \\beta^{-1})\\right)}=\\trace{\\left(\\rho_\\lambda(\\sigma)\\rho_{\\lambda}( \\beta)^{\\dagger}\\right)}\n \\]\n gives\n \\begin{equation}\n \\Pi^L_{\\lambda}|\\psi\\rangle=\\sqrt{\\frac{d_{\\lambda}}{n!}}\\sum_{\\beta\\in S_n}\\trace{\\left(\\hat{f}(\\lambda)\\rho_{\\lambda}(\\beta)^{\\dagger}\\right)}|\\beta\\rangle\\label{eq:povm1}.\n \\end{equation}\n \n Below we show this is equal to $\\mathrm{QFT}_n^{\\dagger}P_\\lambda \\mathrm{QFT}_n|\\psi\\rangle$. We have:\n \\begin{align}\n \\mathrm{QFT}_n^{\\dagger}P_\\lambda \\mathrm{QFT}_n|\\psi\\rangle&= \\mathrm{QFT}_n^{\\dagger}P_\\lambda\\sum_{\\omega\\vdash n}\\sum_{i,j=1}^{d_{\\omega}} (\\hat{f}(\\omega))_{ij}|\\omega,i,j\\rangle \\\\\n &=\\mathrm{QFT}_n^{\\dagger}  \\sum_{i,j=1}^{d_{\\lambda}} (\\hat{f}(\\lambda))_{ij}|\\lambda,i,j\\rangle =\\sqrt{\\frac{d_{\\lambda}}{n!}}\\sum_{\\beta\\in S_n}\\trace{\\left(\\hat{f}(\\lambda)\\rho_{\\lambda}(\\beta)^{\\dagger}\\right)}|\\beta\\rangle,\n \\end{align}\n where we used Eq.~\\eqref{eq:invf} in the last line. This coincides with Eq.~\\eqref{eq:povm1} and completes the proof.\n \\end{proof}\nNext we show that the circuit in Fig.~\\ref{fig:gpe} (b) implements the generalized phase estimation measurement $M_{\\rho}$:\n\\begin{align}\n|\\tau,1,1\\rangle\\otimes |\\psi\\rangle  \\xrightarrow[]{\\mathrm{QFT}_n^{\\dagger}\\otimes I} \\frac{1}{\\sqrt{n!}}&\\sum_{\\alpha\\in S_n}|\\alpha\\rangle \\otimes |\\psi\\rangle \\\\\n\\xrightarrow[]{C-\\rho^{\\dagger}} \n&\\frac{1}{\\sqrt{n!}}\\sum_{\\alpha\\in S_n}|\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle \\\\\n \\xrightarrow[]{\\text{Measure } M_L} \n&\\frac{1}{\\sqrt{n!}}\\sum_{\\alpha\\in S_n}\\Pi^L_{\\omega}|\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle \\quad \\quad (\\text{conditioned on meas. outcome }\\omega)\\\\\n=&\\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle\\\\\n\\xrightarrow[]{C-\\rho} & \\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho(\\sigma \\alpha)\\rho^{\\dagger}(\\alpha)|\\psi\\rangle.\\label{eq:outputstate}\n\\end{align}\nUsing the fact that $\\rho(\\sigma\\alpha)=\\rho(\\sigma)\\rho(\\alpha)$ and that $\\rho^{\\dagger}(\\alpha)=\\rho(\\alpha)^{-1}$ we see that the output state Eq.~\\eqref{eq:outputstate} is equal to\n\\begin{align}\n \\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho(\\sigma)|\\psi\\rangle\n=\\frac{1}{\\sqrt{n!}} \\sum_{\\beta\\in S_n}|\\beta\\rangle \\otimes \\frac{d_\\omega}{n!}\\sum_{\\sigma\\in S_n} \\chi_{\\omega}(\\sigma) \\rho(\\sigma)|\\psi\\rangle.\n\\end{align}\nOn the right hand side the state of the second register is $\\Pi_{\\omega}|\\psi\\rangle$, as desired.\n\n\\section{$\\QMA$, $\\#\\BQP$, and $\\QAPC$}\nRecall the definition of a verification circuit from the main text. In the following we write $p(\\psi)$ for the acceptance probability of a verification circuit with input state $\\psi$. \n\nA (promise) problem is in $\\QMA$ \\textit{with completeness $c$ and soundness $s$}, also denoted $\\QMA(c,s)$, if there exists a uniform polynomial-sized family of verification circuits $C_x$ labeled by instances $x$ of the problem, such that (A) If $x$ is a yes instance then there exists a witness $\\psi$ such that $p(\\psi)\\geq c$, and (B) If $x$ is a no instance then $p(\\psi)\\leq s$ for all  $\\psi$.  \n\nA standard convention is to define $\\QMA\\equiv \\QMA(2\/3,1\/3)$. The definition is not very sensitive to the choice of completeness $c$ and soundness $s$: it is known that $\\QMA(c,s)=\\QMA(2\/3,1\/3)$ whenever $0<s<c\\leq 1$ and $c-s=\\Omega(1\/\\mathrm{poly}(n))$.  In fact, one can reduce the error of any $\\QMA$ verifier exponentially in the input size $n$, so that, e.g.,  $\\QMA(2\/3,1\/3)=\\QMA(4^{-n},1-4^{-n})$ as well \\cite{kitaev2002classical, marriott2005quantum}. \n\nWe choose to define $\\#\\BQP$ as a class of relation problems rather than a class of functions, languages (decision problems), or promise problems. Recall from the main text that Eq.~\\eqref{eq:POVM} describes the POVM element corresponding to the accept outcome of a quantum verification circuit and $N_q$ denotes the number of eigenvalues of $A$ that are at least $q$. We define $\\#\\BQP$ as the class of problems that can be expressed as: given a quantum verification circuit with an $n$-qubit input register, total size $\\mathrm{poly}(n)$, and two thresholds $0<b<a\\leq 1$ and $a-b=\\Omega(1\/\\mathrm{poly}(n))$, compute an estimate $N$ such that $N_a\\leq N\\leq N_b$. This defines a relation problem because there may be more than one solution for a given instance (this happens whenever $N_a\\neq N_b$).\n\n Ref.~\\cite{brown2011computational} puts forward a slightly different definition of $\\#\\BQP$, as the class of promise problems obtained by augmenting and specializing the above definition with the promise that $N_a=N_b$. This has the pleasing feature that it results in a class of functions rather than relations, though at the expense of adding a promise.\n\nOne of the main results established in Ref.~\\cite{brown2011computational} is that any problem in  $\\#\\BQP$ (as they define it) can be efficiently reduced to computing a $\\#\\P$ function. We now describe how their proof  can be straightforwardly adapted to our setting, establishing the same result for the definition of $\\#\\BQP$ that we use in this paper.  Using the in-place error reduction for $\\QMA$ \\cite{marriott2005quantum} we may assume without loss of generality that $a=1-2^{-n-2}$ and $b=2^{-n-2}$ (see, e.g. the proof of Lemma 8 in the supplementary material of Ref.~\\cite{bravyi2022quantum}).\nThen \n\\begin{equation}\n\\mathrm{Tr}(A)\\leq N_b+2^{-n-2}\\cdot (2^n-N_b)\\leq N_b+1\/4,\n\\label{eq:trup}\n\\end{equation}\nand\n\\begin{equation}\n\\mathrm{Tr}(A)\\geq N_a (1-2^{-n-2})\\geq N_a-1\/4.\n\\label{eq:trlow}\n\\end{equation}\nEqs.~(\\ref{eq:trup},\\ref{eq:trlow}) show that the nearest integer $N$ to $\\mathrm{Tr}(A)$ solves the given $\\#\\BQP$ problem as it satisfies $N_a\\leq N\\leq N_b$. It then suffices to show that we can compute $\\mathrm{Tr}(A)$ to within an additive error of $1\/4$ using efficient classical computation along with a single call to a $\\#\\P$ oracle. This is established in Theorem 14 of Ref.~\\cite{brown2011computational} via a proof that is based on expressing $\\mathrm{Tr}(A)$ as a sum over paths, see Ref.~\\cite{brown2011computational} for details.\n\n\n\\section{Efficient sampling algorithm}\nNow let us describe the algorithm stated in Lemma \\ref{lem:samplekron}, which appears implicitly in Ref.~\\cite{Moore07}. \n\nThe completely mixed state over the image of the projector $\\Pi^L_{\\mu}$ can be prepared as follows.\n\\begin{align}\n    \\frac{\\Pi^L_\\mu}{d_\\mu^2} &= \\text{QFT}_n \\left( \\ket{\\mu} \\bra{\\mu} \\otimes I\/d_{\\mu} \\otimes  I\/d_{\\mu}\\right) \\text{QFT}_n^\\dag. \n\\end{align}\nStarting from $\\frac{\\Pi^L_\\mu \\otimes \\Pi^L_\\nu}{d_\\mu^2 d_\\nu^2}$ we then measure the projector $\\Pi^L_\\lambda$ using Weak Fourier sampling. We obtain outcome $\\lambda$ with probability: \n\\begin{align}\n    p(\\lambda) &= \\text{Tr} \\left(\\frac{\\Pi^L_\\mu \\otimes \\Pi^L_\\nu}{d_\\mu^2 d_\\nu^2} \\Pi_\\lambda \\right) = \\frac{d_\\lambda}{d_\\mu d_\\nu n!} \\sum_{g,h,\\ell} \\chi_\\mu(g) \\chi_\\nu(h) \\chi_\\lambda(\\ell)  = \\frac{d_\\lambda}{d_\\mu d_\\nu} g_{\\mu \\nu \\lambda}.\n\\end{align}\n\n\n\\section{Row sums}\n\n\\begin{lemma}\nThe multiplicity of an irreducible representation $\\rho_\\lambda$ in the conjugation representation $\\rho_c$ is given by the row sum $R_{\\lambda}$ defined in Eq.~\\eqref{eq:rowsum}.\n\n\\end{lemma}\n\\begin{proof}\nRecall that the centralizer $Z(\\pi)$ of an element $\\pi \\in S_n$ is defined as the set of elements $g \\in S_n$ that commute with $\\pi$. The conjugation representation acts on $\\mathbb{C}S_n$ as\n\\[\n\\rho_c(\\pi)|\\sigma\\rangle = |\\pi \\sigma \\pi^{-1}\\rangle \n\\]\nand therefore \n\\[\n\\chi_{c}(\\pi)\\equiv \\mathrm{Tr}(\\rho_c(\\pi))=|Z(\\pi)|,\n\\]\nwhere $Z(\\pi)$ is the centralizer of $\\pi \\in S_n$. Note that all elements $\\pi\\in S_n$ in the same conjugacy class $C$ have the same centralizer which we denote $Z(C)$.\n\nWe can decompose the conjugation representation into irreducibles as $\\rho_c \\simeq \\bigoplus_{\\lambda \\vdash n} m_{\\lambda} \\rho_{\\lambda}$ for some multiplicities $m_\\lambda$. Taking the trace gives \n\\[\n\\chi_c(\\pi)=\\sum_{\\lambda \\vdash n} m_{\\lambda}\\chi_{\\lambda}(\\pi)\n\\]\nfor all $\\pi\\in S_n$. Using orthogonality of the irreducible characters we arrive at\n\\begin{align}\n    m_\\lambda &= \\frac{1}{n!} \\sum_{g \\in G} \\chi_\\lambda(g) \\chi_{c}(g) = \\frac{1}{n!} \\sum_{C \\vdash n} \\chi_\\lambda(C) |C| |Z(C)| \\label{eq:rowsumC},\n\\end{align}\nwhere $|C|$ is the order of the conjugacy class $C$. By Lagrange's theorem, $n! \/ |C| = |Z(C)|$, and substituting this into Eq.~\\eqref{eq:rowsumC} shows that $m_{\\lambda}=R_{\\lambda}$ and completes the proof.\n\n\n\\end{proof}\n\\begin{lemma}\n$R_{\\lambda}=d_{\\lambda}^{-1}\\mathrm{Tr}(\\Pi_{\\lambda}^{c})$\n\\end{lemma}\n\\begin{proof}\nBy definition, \n\\[\n\\Pi_{\\lambda}^{c}=\\frac{d_{\\lambda}}{n!}\\sum_{g\\in S_n}\\chi_{\\lambda}(g)\\rho_c(g),\n\\]\nand therefore\n\\[\n\\mathrm{Tr}(\\Pi_{\\lambda}^{c})=\\frac{d_{\\lambda}}{n!}\\sum_{g\\in S_n}\\chi_{\\lambda}(g)\\chi_c(g) = d_\\lambda R_\\lambda,\n\\]\nby Eq. \\eqref{eq:rowsumC}\n\\end{proof}\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Sequential Patterns}\n\nLet $\\I$ be a finite set of {\\em items}. \nThe language of sequences  corresponds to $\\mathcal{L_I}=\\I^n$ where $n\\in\\mbox{$\\mathbb{N}$}^+$. \n\n\\begin{definition}[sequence, sequence database]\nA sequence $s$ over $\\mathcal{L_I}$ is an ordered list\n$\\langle s_1 s_2 \\ldots s_n \\rangle$, where $s_i$, $1 \\leq i \\leq n$,\nis an item. $n$ is called the length of the sequence $s$.\nA sequence database $\\SDB$ is a set of tuples $(sid, s)$,\nwhere $sid$ is a sequence identifier and $s$ a sequence. \n\\end{definition}\n\n\\begin{definition}[subsequence,  $\\mbox{$\\preceq$}$ relation]\nA sequence $\\alpha = \\langle \\alpha_1 \\ldots \\alpha_m \\rangle$ is a \n  subsequence of $s = \\langle s_1 \\ldots s_n \\rangle$,\ndenoted by ($\\alpha$ $\\mbox{$\\preceq$}$ $s$), if $m \\leq n$ and there exist integers \n$1 \\leq j_1 \\leq \\ldots \\leq j_m\\leq n$, such that \n$\\alpha_i=s_{j_i}$ for all $1 \\leq i \\leq m$.  We also say that $\\alpha$ is\ncontained in $s$ or $s$ is a super-sequence of $\\alpha$. \nFor example, the sequence $\\angx{BABC}$ is a super-sequence of\n$\\angx{AC}$ : $\\angx{AC}$ $\\mbox{$\\preceq$}$  $\\angx{BABC}$.\nA tuple $(sid, s)$  contains a sequence $\\alpha$, if $\\alpha$ $\\mbox{$\\preceq$}$ $s$. \n\\end{definition}\n\nThe cover of a sequence $p$ in $\\SDB$ is the set of all tuples in\n$\\SDB$ in which $p$ is contained. \nThe support of a sequence $p$ in $\\SDB$  is the number of \ntuples in $\\SDB$ which contain $p$. \n\n\\begin{definition}[coverage, support]\nLet $\\SDB$ be a sequence database and $p$ a sequence. \n$cover_{\\SDB}(p)$$=$$\\{(sid, s) \\in \\SDB \\, | \\, p \\,\\mbox{$\\preceq$}\\, s\\}$ and\n$sup_{\\SDB}(p)=\\# cover_{\\SDB}(p)$. \n\\end{definition}\n\n\\begin{definition}[sequential pattern]\nGiven a minimum support threshold $minsup$, every sequence $p$ \nsuch that $sup_{\\SDB}(p) \\geq minsup$ is called a sequential pattern~\\cite{DBLP:conf\/icde\/AgrawalS95}. \n$p$ is said to be frequent in $\\SDB$. \n\\end{definition}\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\t\\scalebox{0.90}{\n\t\t\t\\begin{tabular}{|l|l|}\t\n\t\t\t\t\\hline\n                                ~sid~ &~Sequence~ \\\\\n\t\t\t\t\\hline\n\t\t\t\t $1$ & $\\angx{ABCBC}$  \\\\\n\t\t\t\t $2$ & $ \\angx{BABC}$ \\\\\n\t\t\t\t $3$ & $ \\angx{AB} $ \\\\\n\t\t\t\t $4$  & $\\angx{BCD}$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular} \n\t\t}  \n                        \\caption{$\\SDB_1$: a sequence database example.}\n\t\t\\label{tab:SDB}\n\t\\end{center}\n\\end{table}\t\n\n\\begin{example}\nTable~\\ref{tab:SDB} represents\n  a sequence database of four sequences \nwhere the set of items is $ \\I = \\{A, B, C, D\\}$. Let \nthe sequence $p= \\angx{A C}$. We have\n$cover_{{\\SDB}1}(p)=\\{(1, s_1), (2, s_2)\\}$. If we consider \n$minsup=2$, $p=\\angx{A C}$ is a sequential pattern because $sup_{\\SDB_1}(p) \\ge 2$. \n\\end{example}\n\n\\begin{definition}[sequential pattern mining (SPM)]\nGiven a sequence database $\\SDB$ and a minimum support threshold\n$minsup$. The problem of sequential pattern mining is to find all\npatterns $p$ such that $sup_{\\SDB}(p) \\geq minsup$. \n\\end{definition}\n\n\n\\subsection{SPM under Constraints}\n\\label{sec:local}\n\nIn this section, we define the problem of mining sequential \npatterns in a sequence database satisfying user-defined\nconstraints Then, we review the most usual constraints for\nthe sequential mining problem~\\cite{Pei2002}. \n\n\\noindent \n{\\bf Problem statement}. Given a constraint $C(p)$ on \npattern $p$ and a sequence database \\SDB, the problem\nof constraint-based pattern mining is to find the\ncomplete set of patterns satisfying $C(p)$.\nIn the following, we present different types of constraints\nthat we explicit in the context of sequence mining. All\nthese constraints will be handled by our concise encoding (see\nSections \\ref{consistency} and \\ref{pattern-encoding}). \n\n\\noindent\n- The minimum size constraint $size(p,\\ell_{min})$ states that \nthe number of items of $p$ must be greater than or equal to $\\ell_{min}$. \n\n\\noindent\n- The item constraint $item(p,t)$ states that an item $t$ must belong\n(or not) to a pattern $p$. \n\n\\noindent\n- The regular expression\nconstraint~\\cite{DBLP:journals\/tkde\/GarofalakisRS02}\n$reg(p,exp)$ states that a pattern $p$ must be \naccepted by the deterministic finite \nautomata associated to the regular expression $exp$. \n\n\n\\subsection{Projected Databases}\n\\label{PB-projected-databases}\nWe now present the necessary definitions related to the concept of\n{\\it projected databases}~\\cite{DBLP:conf\/icde\/PeiHPCDH01}.\n\n\\begin{definition}[prefix, projection, suffix]  \nLet $\\beta=\\angx{\\beta_1 \\ldots \\beta_n}$ and $\\alpha = \\angx{\\alpha_1\n  \\ldots \\alpha_m}$ be two sequences, where $m \\leq n$. \\\\ \n- Sequence $\\alpha$ is called the prefix of $\\beta$ iff $\\forall i \\in\n[1..m], \\alpha_i=\\beta_i$.   \\\\ \n- Sequence $\\beta=\\angx{\\beta_1 \\ldots \\beta_n}$ is called the \nprojection of some sequence $s$ w.r.t. $\\alpha$,  \niff (1) $\\beta\\, \\mbox{$\\preceq$}\\, s$, \n(2) $\\alpha$ is a prefix of $\\beta$ \nand (3) there exists no proper super-sequence $\\beta'$ of $\\beta$ such\nthat $\\beta' \\,\\mbox{$\\preceq$}\\,s$ and $\\beta'$ also has $\\alpha$ as prefix. \\\\\n- Sequence $\\gamma =\\angx{\\beta_{m+1} \\ldots \\beta_n}$ is called the suffix of $s$\nw.r.t. $\\alpha$. With the standard concatenation operator \"$concat$\",\nwe have $\\beta = concat(\\alpha, \\gamma)$.  \n\\end{definition}\n\n\\begin{definition}[projected database] \nLet  $\\SDB$ be a sequence database, \nthe $\\alpha$-projected database, denoted by $SDB|_\\alpha$, \nis the collection of suffixes of sequences in  $\\SDB$ w.r.t. prefix $\\alpha$. \n\\end{definition}\n\n\\cite{DBLP:conf\/icde\/PeiHPCDH01} have proposed an\nefficient algorithm, called \\code{PrefixSpan}, for mining sequential\npatterns based on the concept of {\\it projected databases}.   \nIt proceeds by dividing the initial database into smaller ones\nprojected on the frequent subsequences obtained so far; only their\ncorresponding suffixes are kept. Then, sequential patterns are mined \nin each projected database by exploring only locally  frequent\npatterns. \n\n\\begin{example}\n\\label{exp-projection}\nLet us consider the sequence database of Table~\\ref{tab:SDB} with\n$minsup=2$.  \n\\code{PrefixSpan} starts by scanning ${\\SDB}_1$ to find\nall the frequent items, each of them is used as\na prefix to get projected databases. \nFor ${\\SDB}_1$, we get $3$ disjoint subsets w.r.t. the prefixes\n$\\angx{A}$, $\\angx{B}$, and $\\angx{C}$.  \nFor instance, ${\\SDB}_1|_{\\angx{A}}$ consists of 3 suffix sequences:\n$\\{(1,\\angx{B C B C})$, $(2,\\angx{BC})$, $(3,\\angx{B})\\}$.   \nConsider the projected database ${\\SDB}_1|_{<A>}$, its\nlocally frequent items are $B$ and $C$. Thus, ${\\SDB}_1|_{<A>}$\ncan be recursively partitioned into 2 subsets w.r.t. the two \nprefixes $\\angx{A B}$ and $\\angx{A C}$. The $\\angx{A B}$- and $\\angx{A\n  C}$- projected databases can be constructed and recursively mined\nsimilarly. The processing of a \n$\\alpha$-projected database terminates  \nwhen no frequent subsequence can be generated. \n\\end{example}\n\nProposition~\\ref{prop-prefixspan} establishes the support count of a\nsequence $\\gamma$ in $\\SDB|_\\alpha$~\\cite{DBLP:conf\/icde\/PeiHPCDH01}: \n\\begin{proposition}[Support count]\\label{prop-prefixspan}\nFor any sequence $\\gamma$ in $\\SDB$ with prefix $\\alpha$ and suffix $\\beta$\ns.t. $\\gamma= concat(\\alpha,\\beta$), $sup_{\\SDB}(\\gamma) = sup_{\\SDB|_{\\alpha}}(\\beta)$. \n\\end{proposition}\n\nThis proposition ensures that only the sequences in $\\SDB$ grown from\n$\\alpha$ need to be considered for the support count of a sequence\n$\\gamma$. Furthermore, only those suffixes with prefix \n$\\alpha$ should be counted. \n\n\n\\subsection{CSP and Global Constraints}\n\n\\noindent\nA {\\it Constraint Satisfaction Problem} (CSP) consists of a \nset $X$ of $n$ variables, a domain $\\mathcal{D}$ mapping each variable \n$X_i \\in X$ to a finite set of values $D(X_i)$, and a set of constraints\n$\\mathcal{C}$. An assignment $\\sigma$ is a mapping from variables in $X$ to\nvalues in their domains: $\\forall X_i \\in X, \\sigma(X_i) \\in D(X_i)$. \nA constraint $c \\in \\mathcal{C}$ is a subset of the \ncartesian product of the domains of the variables that are in $c$. \nThe goal is to find an assignment \nsuch that all constraints are satisfied.  \n\n\\noindent \n\\textbf{Domain consistency (DC).} \nConstraint solvers typically use backtracking search to explore the\nspace of partial assignments. At each assignment, filtering\nalgorithms prune the search space by enforcing local consistency\nproperties like domain consistency. A constraint $c$ on $X$ is domain\nconsistent, if and only if, for every $X_i \\in X$ and for every\n$d_i \\in D(X_i)$, there is an assignment $\\sigma$ satisfying $c$ such\nthat $\\sigma(X_i) = d_i$. Such \nan assignment is called a support.  \n\n\\noindent \n\\textbf{Global constraints} \nprovide shorthands to often-used combinatorial substructures.   \nWe present two global constraints.\nLet $X=\\angx{X_1, X_2, ..., X_n}$ be a sequence of $n$ variables.\n\n\\noindent\nLet $V$ be a set of values, $l$ and $u$ be two integers s.t. $0\\leq l \\leq u \\leq n$,\nthe constraint \\texttt{Among}$(X, V ,l, u)$ states that each value $a \\in V$ \nshould occur at least $l$ times and at most $u$ times in $X$~\\cite{DBLP:journals\/jmcm\/Beldi94}.\nGiven a deterministic finite automaton $A$,\nthe constraint \\texttt{Regular}$(X, A)$ ensures that the sequence $X$\nis accepted by $A$~\\cite{regular}.  \n\n\n\n\\subsection{Ad hoc Methods for SPM}\n{\\tt GSP}~\\cite{DBLP:conf\/edbt\/SrikantA96} was the first algorithm\nproposed to extract sequential patterns. It uses a generate-and test\napproach. Later, two major classes of methods have been proposed: \n\n\\noindent -  Depth-first search based on a vertical database format\ne.g. \\texttt{cSpade}\\xspace incorporating contraints (max-gap, max-span,\nlength)~\\cite{DBLP:conf\/cikm\/Zaki00}, \n\\texttt{SPADE}\\xspace~\\cite{DBLP:journals\/ml\/Zaki01}   \n or {\\tt\n   SPAM}~\\cite{Ayres:2002:SPM:775047.775109}.\n\n\\noindent - Projected pattern growth such as {\\tt\n  PrefixSpan}~\\cite{DBLP:conf\/icde\/PeiHPCDH01}  \nand its extensions, e.g. \\texttt{CloSpan}\\xspace\\ for mining closed sequential\npatterns~\\cite{DBLP:conf\/sdm\/YanHA03} or \\texttt{Gap-BIDE}\\xspace~\\cite{Li2012}\ntackling the gap constraint. \n\nIn \\cite{DBLP:journals\/tkde\/GarofalakisRS02}, the authors\n  proposed {\\tt SPIRIT} based on {\\tt GSP}\n  for SPM with regular expressions. Later, \\cite{Bonchi:2008}\n  introduces Sequence Mining Automata (\\sma), a new approach\n  based on a specialized kind of Petri Net. Two variants of \\sma\n  were proposed:  \\smap (\\sma one pass) and \\texttt{SMA-FC}\\xspace (\\sma Full\n  Check).  \\smap processes by means\n  of the \\sma all sequences one by one, and enters all resulting valid\n  patterns in a hash table for support counting, while \\texttt{SMA-FC}\\xspace allows\n  frequency based pruning during the scan of the database. \nFinally, \\cite{Pei2002} provides a survey for other constraints \nsuch as regular expressions, length and aggregates. \nBut, \nall these proposals, though efficient, are ad hoc methods suffering\nfrom a lack of genericity. \nAdding new constraints often requires to develop new implementations.\n\n\n\\subsection{CP Methods for SPM}\n\\label{CP4SPM}\n\nFollowing the work of \\cite{DBLP:journals\/ai\/GunsNR11} for itemset mining,\nseveral methods have been proposed to mine sequential patterns using CP.\n\n\\noindent {\\bf Proposals.}\n\\cite{DBLP:conf\/ecai\/CoqueryJSS12} have\nproposed a first SAT-based model for discovering a special class of\npatterns with wildcards\\footnote{A wildcard is a special symbol that\n  matches any item of $\\I$ including itself.} in a single\nsequence under different types of \nconstraints (e.g. frequency, maximality, closedness). \n{\\cite{metivierLML13} have proposed a CSP model for SPM.\nEach sequence is encoded by an automaton capturing all subsequences that can occur in it.\n\\cite{DBLP:conf\/ictai\/KemmarULCLBC14} have proposed a CSP model for SPM with wildcards.\nThey show how some constraints dealing with local patterns (e.g. frequency, size, gap, regular expressions)\nand constraints defining more complex patterns such as relevant subgroups \\cite{DBLP:journals\/jmlr\/NovakLW09} \nand \\topk patterns can be modeled using a CSP. \n\\cite{NegrevergneCPIAOR15} have proposed two CP encodings for the SPM. \nThe first one uses a global constraint to encode the subsequence relation (denoted \\texttt{global-p.f}\\xspace),\nwhile the second one encodes explicitly this relation using additional\nvariables and constraints (denoted \\texttt{decomposed-p.f}\\xspace).\n\nAll \nthese proposals use {\\bf reified constraints} to encode the database.\nA reified constraint associates a boolean variable to a constraint\nreflecting whether the constraint is satisfied (value 1) or not (value\n0).   \nFor each sequence $s$ of $\\SDB$, a reified constraint, \nstating whether (or not) the unknown pattern $p$ is a subsequence of $s$,\nis imposed: $(S_s=1) \\Leftrightarrow (p \\preceq s)$.\nA great consequence is that the encoding of the frequency measure \nis straightforward: $freq(p) = \\sum_{s \\in \\SDB} S_s$.\nBut such an encoding has a major drawback since it\nrequires $(m=\\#\\SDB)$ reified constraints to encode the whole\ndatabase. This constitutes a strong limitation of the size of the\ndatabases that could be managed. \n\nMost\nof these proposals encode {\\bf the subsequence relation} $(p \\preceq s)$ \nusing variables $Pos_{s,j}$ $(s \\in  \\SDB$ and $1 \\le j \\le \\ell)$\nto determine a position where $p$ occurs in $s$.\nSuch an encoding requires a large number of additional \nvariables ($m$$\\times$$\\ell$) and makes the labeling computationally\nexpensive. \nIn order to address this drawback, \\cite{NegrevergneCPIAOR15} have\nproposed a global constraint \\texttt{exists-embedding}  \nto encode the subsequence relation,\nand used projected frequency within an ad hoc specific branching strategy\nto keep only frequent  items before branching over the variables of\nthe pattern. \nBut, \nthis encoding still relies on reified constraints and requires to\nimpose $m$ \\texttt{exists-embedding} global constraints. \n\n\nSo,\nwe propose in the next section the \\textsc{Prefix-Projection}\\xspace global constraint \nthat  fully exploits the principle of projected databases to encode\nboth the subsequence relation and the frequency constraint. \n \\textsc{Prefix-Projection}\\xspace does not require any reified constraints nor any extra\n variables to encode the subsequence relation. \nAs a consequence, usual SPM constraints (see Section~\\ref{sec:local}) \ncan be encoded in a straightforward way using directly the (global)\nconstraints of the CP solver. \n\n\n\n\n\n\\subsection{A Concise Encoding}\n\\label{sec:pattern}\n\nLet $P$ be the unknown pattern of size $\\ell$ we are looking for. \nThe symbol $\\vide$ stands for an empty item and denotes the end of a\nsequence.  \nThe unknown pattern $P$ is encoded with a sequence\nof $\\ell$ variables $\\angx{P_1,P_2,\\ldots,P_\\ell}$\ns.t. $\\forall i \\in [1\\ldots\\ell],  D(P_ i)= \\I\\cup\\{\\vide\\}$.  \nThere are two basic rules on the domains:\n\\begin{enumerate}\n\\vspace*{-.1cm}\\item \nTo avoid the empty sequence, the first item of $P$ must be non empty,\nso $(\\vide\\not\\in D_1)$.  \n\\item\nTo allow patterns with less than $\\ell$ items, we impose that $\\forall\ni \\in [1.. (\\ell$$-$$1)], (P_i=\\vide) \\rightarrow (P_{i+1} = \\vide)$. \n\\end{enumerate}\n\n\\subsection{Definition and Consistency Checking} \n\\label{consistency}\n\nThe global constraint \\textsc{Prefix-Projection}\\xspace ensures both subsequence relation and\nminimum frequency constraint.  \n\n\\begin{definition}[\\textsc{Prefix-Projection}\\xspace global constraint]\nLet $P = \\angx{P_1,P_2,\\ldots,P_\\ell}$ be a pattern of size $\\ell$.   \n$\\angx{d_1, ..., d_{\\ell}} \\in D(P_1)\\times \\ldots \\times D(P_\\ell)$\nis a solution of \\textsc{Prefix-Projection}\\xspace$(P, \\SDB, minsup)$ iff   \n$sup_{\\SDB}(\\angx{d_1, ..., d_{\\ell}})\n\\geq minsup$. \n\\end{definition}\n\n\\begin{proposition}\n\\label{prop-solution}\nA \\textsc{Prefix-Projection}\\xspace$(P, \\SDB,minsup)$ constraint has a solution if and only if  \nthere exists an assignment $\\sigma = \\angx{d_1, ..., d_{\\ell}}$ of\nvariables of $P$ s.t. $\\SDB|_{\\sigma}$ has at least $minsup$ suffixes of $\\sigma$: \n$\\#\\SDB|_{\\sigma}\\geq minsup$. \n\\end{proposition}\n\n{\\it Proof: }\nThis is a direct consequence of proposition \\ref{prop-prefixspan}. We\nhave straightforwardly   \n$sup_{\\SDB}(\\sigma) = sup_{\\SDB|_{\\sigma}}(\\angx{}) =\n\\#\\SDB|_{\\sigma}$. Thus, suffixes of $\\SDB|_{\\sigma}$ are supports  \nof $\\sigma$ in the constraint \\textsc{Prefix-Projection}\\xspace$(P, \\SDB,minsup)$,\nprovided that $\\#\\SDB|_{\\sigma}\\geq minsup$. \n$\\Box$\n\nThe following proposition characterizes values in the domain\nof unassigned (i.e. future) variable $P_{i+1}$ that are consistent with the \ncurrent assignment of variables $\\angx{P_1, ..., P_i}$.\n\n\\begin{proposition}\n\\label{prop-consistency}\nLet $\\sigma \\footnote{We indifferently denote $\\sigma$ by $\\angx{d_1,\n    \\dots, d_i}$ or by $\\angx{\\sigma(P_1), \\dots, \\sigma(P_{i})}$.} =\\angx{d_1, \\dots, d_i}$ be a current\nassignment of\nvariables $\\angx{P_1, \\dots, P_i}$, $P_{i+1} $ be a future variable. A\nvalue $d \\in D(P_{i+1})$ \nappears in a solution for \\textsc{Prefix-Projection}\\xspace$(P, \\SDB, minsup)$ if and only if\n$d$ is a frequent item in $\\SDB|_{\\sigma}$:  \n$$\\#\\{(sid,\\gamma) | (sid,\\gamma) \\in \\SDB|_{\\sigma} \\wedge \\angx{d}\\mbox{$\\preceq$}\n\\gamma\\} \\geq minsup$$\n\\end{proposition}\n\n\\noindent\n{\\it Proof: }\nSuppose that value $d \\in D(P_{i+1})$ occurs in $\\SDB|_{\\sigma}$\nmore than $minsup$. From proposition~\\ref{prop-prefixspan}, we have \n$sup_{\\SDB}(concat(\\sigma, \\angx{d})) =\nsup_{\\SDB|_{\\sigma}}(\\angx{d})$. Hence, the\nassignment $\\sigma \\cup \\angx{d}$ satisfies the constraint,\nso $d \\in D(P_{i+1})$ participates in a solution. \n$\\Box$\n\n\\noindent {\\bf Anti-monotonicity of the frequency measure.} \nIf a pattern $p$ is not frequent, then any pattern $p'$ satisfying \n$p \\,\\mbox{$\\preceq$}\\, p'$ is not frequent.   \nFrom proposition~\\ref{prop-consistency} and according \nto the {\\it anti-monotonicity property}, we can derive the following \npruning rule: \n\n\\begin{proposition}\n\\label{prop-filtering}\nLet $\\sigma=\\angx{d_1, \\dots, d_i}$ be a current assignment of\nvariables $\\angx{P_1, \\dots, P_i}$. All values $d \\in D(P_{i+1})$\nthat are locally not frequent in\n$\\SDB|_{\\sigma}$ can be pruned from the domain of variable\n$P_{i+1}$. Moreover, these values $d$ can also be pruned from the domains\nof variables $P_j$ with  $j \\in [i+2, \\dots, \\ell]$. \n\\end{proposition}\n\n\\noindent\n{\\it Proof: } \nLet $\\sigma=\\angx{d_1, \\dots, d_i}$ be a current assignment of\nvariables $\\angx{P_1, \\dots, P_i}$. Let $d \\in D(P_{i+1})$ s.t. \n$\\sigma' = {concat(\\sigma,\\angx{d})}$. \nSuppose that $d$ is not frequent in $\\SDB|_{\\sigma}$. \nAccording to proposition \\ref{prop-prefixspan},\n$sup_{\\SDB|_{\\sigma}}(\\angx{d}) = sup_{\\SDB}(\\sigma')<minsup$, thus $\\sigma'$\nis not frequent. So, $d$ can be pruned from the domain of $P_{i+1}$.\n\n\\noindent\nSuppose that the assignment $\\sigma$ has been extended to\n$concat(\\sigma, \\alpha)$, where $\\alpha$ corresponds to the assignment\nof variables $P_j$ (with $j > i$).  \nIf $d \\in D(P_{i+1})$ is not frequent, it is straightforward that\n$sup_{\\SDB|_{\\sigma}}(concat(\\alpha, \\angx{d}))\\leq\nsup_{\\SDB|_{\\sigma}}(\\angx{d}) < minsup$. Thus, if $d$ is not frequent\nin $\\SDB|_{\\sigma}$, it will be also not frequent in\n$\\SDB|_{concat(\\sigma,\\alpha)}$. \nSo, $d$ can be pruned from the domains of\n$P_j$ with $j \\in [i+2, \\dots, \\ell]$.\n$\\Box$\n\n\\begin{example}\nConsider the sequence database of Table~\\ref{tab:SDB} with\n$minsup=2$. Let $P = \\angx{P_1,P_2,P_3}$ with $D(P_1) = \\I$ and \n$D(P_2) = D(P_3) = \\I\\cup\\{\\vide\\}$. Suppose that $\\sigma(P_1) = A$, \n$\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ will remove values $A$ and\n$D$ from $D(P_2)$ and $D(P_3)$, since the only locally frequent items in\n${\\SDB}_1|_{<A>}$ are $B$ and $C$.\n\\end{example} \n\nProposition~\\ref{prop-filtering} guarantees \nthat any value (i.e. item)  \n$d \\in D(P_{i+1})$ present but not frequent in $\\SDB|_{\\sigma}$ does not need\nto be considered when extending $\\sigma$, thus avoiding searching over\nit. \nClearly, our global constraint encodes the anti-monotonicity\nof the frequency measure in a simple and elegant way, while \nCP methods for SPM have difficulties to handle this property. \nIn \\cite{NegrevergneCPIAOR15}, this is achieved by using very specific\npropagators and branching  \nstrategies, making the integration quite complex (see\n\\cite{NegrevergneCPIAOR15}). \n\n\n\\begin{algorithm}[t]\n\\begin{small}\n\\caption{\\small \\codex{ProjectSDB}($\\SDB$, $ProjSDB$, $\\alpha$)  \\label{algo:projection}}\n\\KwData{$\\SDB$: initial database; $ProjSDB$: projected sequences; $\\alpha$: prefix}\n\\Begin\n{\n\\lnl{p1}$\\SDB|_{\\alpha} \\leftarrow \\emptyset$ \\;\n\\lnl{p2}\\For{each pair $(sid,start) \\in ProjSDB$}{\n\\lnl{p3}$s \\leftarrow \\SDB[sid]$ \\; \n\\lnl{p4}$pos_{\\alpha}  \\leftarrow 1$;\\ $pos_s  \\leftarrow start$ \\;\n\\lnl{p6}\\While{$(pos_{\\alpha} \\leq \\#\\alpha$ $\\wedge$ $pos_s \\leq \\# s)$}{\n\\lnl{p7}\\If{$(\\alpha[pos_\\alpha] = s[pos_s])$}{\n\\lnl{p8}$pos_\\alpha \\leftarrow pos_\\alpha + 1$ \\;\n}\n\\lnl{p9}$pos_s \\leftarrow pos_s + 1$ \\;\n\n}\n\\lnl{p10}\\If{$(pos_\\alpha = \\#\\alpha + 1)$}{\n\\lnl{p11}$\\SDB|_{\\alpha} \\leftarrow \\SDB|_{\\alpha} \\cup \\{(sid, pos_s)\\}$\n}\n}\n\\lnl{p12}\\Return $\\SDB|_{\\alpha}$ \\;\n}\n\\end{small}\n\\end{algorithm}\n\n\n\\subsection{Building the projected databases.} \n\\label{sec-ProjectSDB}\nThe key issue of our approach lies in the construction of the \nprojected databases. \nWhen projecting a prefix, instead of storing the whole suffix as a\nprojected subsequence, one \ncan represent each suffix by a pair $(sid, start)$ where $sid$ is the\nsequence identifier and $start$ is the starting position of the\nprojected suffix in the \nsequence $sid$. For instance, let us consider the sequence database of Table~\\ref{tab:SDB}. As\nshown in example \\ref{exp-projection}, ${\\SDB}|_{\\angx{A}}$\nconsists of 3 suffix sequences: $\\{(1,\\angx{B C \n  B C})$, $(2,\\angx{BC})$, $(3,\\angx{B})\\}$. \nBy using the {\\it pseudo-projection}, ${\\SDB}|_{\\angx{A}}$ can be\nrepresented by the following three pairs: $\\{(1,2)$, $(2,3)$, $(3,2)\\}$. \nThis is the principle of {\\it pseudo-projection}, adopted in \\texttt{PrefixSpan}\\xspace, \nexploited during the filtering step of our \\textsc{Prefix-Projection}\\xspace global\nconstraint. Algorithm \\ref{algo:projection} details \nthis principle. It takes as input a set of projected sequences\n$ProjSDB$ and a prefix $\\alpha$.  \nThe algorithm processes all the pairs $(sid,start)$ of $ProjSDB$ one\nby one (line \\ref{p2}), and searches for the lowest location of\n$\\alpha$ in the sequence $s$ corresponding to the $sid$ of that\nsequence in $\\SDB$ (lines \\ref{p7}-\\ref{p9}). \n\nIn the worst case, \\codex{ProjectSDB} processes all the items of all\nsequences. So, the time complexity is $O(\\ell\\times m)$, with  \n$m=\\#\\SDB$ and $\\ell$ is the length of the longest sequence in \n$\\SDB$. \nThe worst case space complexity of pseudo-projection is $O(m)$, since\nwe need to store for each sequence only a pair ($sid, start$), while\nfor the standard projection the space complexity is $O(m\\times\n\\ell)$. Clearly, the pseudo-projection takes much less space than the standard projection.\n\n\\begin{algorithm}[!ht]\n\\begin{small}\n\\SetKwFunction{ProcProjection}{\\codex{Function ProjectSDB}}\n\\SetKwFunction{ProcGetFreqItems}{\\codex{Function getFreqItems}}\n\\caption{\\small \\codex{Filter-Prefix-Projection}($\\SDB$, $\\sigma$, $i$, $P$, $minsup$) \\label{algo:filter}}\n\\KwData{$\\SDB$: initial database; $\\sigma$: current prefix\n  $\\angx{\\sigma(P_1), \\ldots,\\sigma(P_i)}$; $minsup$: the minimum\n  support threshold; $\\mathcal{PSDB}$: internal data structure of\n  \\textsc{Prefix-Projection}\\xspace for storing pseudo-projected databases} \n\\Begin\n{\n\\lnl{pre1}\\If{$( i \\geq 2 \\wedge \\sigma(P_i) = \\vide)$}{ \n                  \\lnl{pre2}\\For{$j \\leftarrow i+1$ \\KwTo $\\ell$}{\n                                  \\lnl{pre3}$P_{j}  \\leftarrow \\vide$\\;\n                                 }\n                   \\lnl{pre4} \\Return True\\;\n                   }\n\\Else{\n          \\lnl{pre0} $\\mathcal{PSDB}_ {i} \\leftarrow\n          \\codex{ProjectSDB}(\\SDB, \\mathcal{PSDB}_{i-1},\n          \\angx{\\sigma(P_{i})})$\\;\n          \\lnl{proj5}\\If{$(\\#\\mathcal{PSDB}_{i} < minsup)$}{\n            \\lnl{proj6} \\Return False \\;\n          }\n          \\Else{\n                    \\lnl{proj7} $\\mathcal{FI} \\leftarrow\n                    \\codex{getFreqItems}(\\SDB, \\mathcal{PSDB}_{i}, minsup)$ \\;\n                     \\lnl{proj8}\\For{$j \\leftarrow i+1$ \\KwTo $\\ell$}{\n                                 \\lnl{proj9} \\ForEach{$a \\in D(P_j) \\, s.t. (a \\neq \\vide \\wedge a \\notin \\mathcal{FI} )$}{\n                                   \\lnl{proj10}$D(P_{j}) \\leftarrow\n                                   D(P_{j}) - \\{a\\}$\\;\n                                 }\n                                  }\n                       \\lnl{proj13} \\Return True\\;\n          }\n     }\n}\n\n\n\\ProcGetFreqItems($\\SDB$, $ProjSDB$, $minsup$) \\;\n \\KwData{$SDB$: the initial database; $ProjSDB$: pseudo-projected\n   database; $minsup$: the minimum support threshold; \n   $ExistsItem$, $SupCount$: internal data structures using a hash table\n   for support counting over items;}\n\\Begin\n{\n\\lnl{F1} $SupCount[ ] \\leftarrow \\{0, ..., 0\\}$;\\ \\ $F\\leftarrow \\emptyset$ \\;\n\\lnl{F2}\\For{each pair $(sid,start) \\in ProjSDB$}{\n\\lnl{F3} $ExistsItem[] \\leftarrow \\{false, ..., false\\}$; $s \\leftarrow SDB[sid]$ \\;\n\\lnl{F4}\\For{$i \\leftarrow start$ \\KwTo $\\# s$}{\n\\lnl{F5}$a \\leftarrow s[i]$ \\;\n\n\\lnl{F6}\\If{$(\\neg ExistsItem[a])$}{\n\\lnl{F7}$SupCount[a] \\leftarrow SupCount[a] + 1$ \\;\n \\lnl{F8}$ExistsItem[a] \\leftarrow true$\\;\n\\lnl{F10}\\If{$(SupCount[a] \\geq minsup)$}{\n\\lnl{F11}$F \\leftarrow F \\cup \\{a\\}$\\;\n}\n}\n}\n}\n\\lnl{F12}\\Return $F$\\;\n}\n\\end{small}\n\\end{algorithm}\n\n\n\\subsection{Filtering}\n\\label{Filtering}\n\nEnsuring DC on $\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ is equivalent to finding a sequential pattern of length $(\\ell-1)$ and \nthen checking whether this pattern remains a frequent pattern \nwhen extended to any item $d_{\\ell}$ in $D(P_{\\ell})$. \nThus, finding such an assignment (i.e. support) is as much as difficult than the original \nproblem of sequential pattern mining. \n\\cite{Guizhen-2006} has proved that the problem of \ncounting the number of maximal\\footnote{A sequential pattern $p$ is\nmaximal if there is no sequential pattern $q$ such that $p \\mbox{$\\preceq$} q$.}\nfrequent patterns in a database of sequences is \\#P-complete, thereby\nproving the NP-hardness of \nthe problem of mining maximal frequent sequences. \nThe difficulty is due to the exponential\nnumber of candidates that should \nbe parsed to find the frequent patterns. Thus, finding, for every\nvariable $P_i \\in\nP$ and for every $d_i \\in D(P_i)$, an assignment $\\sigma$ satisfying\n$\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ s.t. $\\sigma(P_i) = d_i$ is of \nexponential nature.  \n\nSo, the filtering of the \\textsc{Prefix-Projection}\\xspace constraint maintains a consistency lower than DC.\nThis consistency is based on specific properties of the projected\ndatabases (see Proposition~\\ref{prop-consistency}), and\nanti-monotonicity of the frequency constraint (see\nProposition~\\ref{prop-filtering}), \nand resembles forward-checking regarding Proposition~\\ref{prop-consistency}. \n\\textsc{Prefix-Projection}\\xspace is considered as a global constraint, since all variables\nshare the same internal data structures that awake and drive the\nfiltering. \n\nAlgorithm~\\ref{algo:filter} describes the  pseudo-code of the\nfiltering algorithm of the \\textsc{Prefix-Projection}\\xspace constraint.  \nIt is an incremental filtering algorithm\nthat should be run when some $i$ first variables are\nassigned according to the following lexicographic ordering $\\angx{P_1,\n  P_2, \\dots, P_\\ell}$ of variables of $P$. \nIt exploits internal data-structures enabling to enhance the\nfiltering algorithm. \nMore precisely, it uses an incremental data structure, denoted\n$\\mathcal{PSDB}$, that stores the intermediate pseudo-projections of\n$\\SDB$, where $\\mathcal{PSDB}_i$ ($i\\in [0, \\ldots, \\ell]$)\ncorresponds to the $\\sigma$-projected \ndatabase of the current partial assignment $\\sigma=\\angx{\\sigma(P_1),\n  \\ldots, \\sigma(P_i)}$ (also called prefix) of variables $\\angx{P_1,\n  \\dots, P_i}$, and $\\mathcal{PSDB}_0 = \\{(sid, 1)| (sid, s) \\in\n\\SDB\\}$ is the initial pseudo-projected database of $\\SDB$ \n(case where $\\sigma = \\angx{}$). \nIt also uses a hash table indexing the items $\\I$ into integers\n$(1 \\ldots \\#\\I)$ for an efficient support counting over items\n(see function {\\tt getFreqItems}).  \n\nAlgorithm~\\ref{algo:filter} takes as input the current partial\nassignment $\\sigma=\\angx{\\sigma(P_1), \\ldots, \\sigma(P_i)}$\nof variables $\\angx{P_1, \\dots, P_i}$, \nthe length $i$ of $\\sigma$ (i.e. position of the last assigned\nvariable in $P$) and the minimum support threshold\n$minsup$. \nIt starts by checking if the last assigned variable\n$P_i$ is instantiated to $\\vide$ (line \\ref{pre1}). In this case, the end\nof sequence is reached (since value $\\vide$ can only appear at the\nend) and the sequence $\\angx{\\sigma(P_1), \\dots, \\sigma(P_{i})}$\nconstitutes a frequent pattern in $\\SDB$; hence the algorithm sets the \nremaining $(\\ell - i)$ unassigned variables to $\\vide$ and returns\n{\\it true}  (lines \\ref{pre2}-\\ref{pre4}). Otherwise, the algorithm\ncomputes incrementally $\\mathcal{PSDB}_i$ from $\\mathcal{PSDB}_{i-1}$ \nby calling function \\codex{ProjectSDB} (see\nAlgorithm~\\ref{algo:projection}).  Then, it checks in line\n\\ref{proj5} whether the current assignment $\\sigma$ is a {\\it legal} prefix\nfor the constraint (see Proposition \\ref{prop-solution}). This is\ndone by computing the size of $\\mathcal{PSDB}_i$. If this size is less\nthan $minsup$, we stop growing $\\sigma$ and we return {\\it\n  false}. \nOtherwise, the algorithm computes the set of\n  locally frequent items $\\mathcal{F_I}$ in $\\mathcal{PSDB}_i$\nby calling function {\\tt getFreqItems} (line\n\\ref{proj7}). \n\n\nFunction  {\\tt getFreqItems} processes all the entries of the\npseudo-projected database \none by one, counts the number of first occurrences of items\n$a$ (i.e. $SupCount[a]$) in each entry $(sid, start)$, and keeps only\nthe frequent ones (lines \\ref{F1}-\\ref{F11}). \nThis is done by using $ExistsItem$ data structure. \nAfter the whole pseudo-projected database has been processed, the\nfrequent items are returned (line \\ref{F12}), and Algorithm~\\ref{algo:filter} \nupdates the current domains of variables $P_j$ with $j \\geq \n(i+1)$ by pruning inconsistent values, \nthus avoiding searching over not frequent items (lines\n\\ref{proj8}-\\ref{proj10}). \n\n\\begin{proposition}\\label{prop-complexity}\nIn the worst case, filtering with \\textsc{Prefix-Projection}\\xspace global constraint can be\nachieved in $O(m\\times\\ell + m\\times d + \\ell\\times d)$. The worst\ncase space complexity of \\textsc{Prefix-Projection}\\xspace is $O(m\\times \\ell)$. \n\\end{proposition}\n\n\\noindent\n{\\it Proof: }\nLet $\\ell$ be the length of the longest sequence in $\\SDB$, $m$ $=$\n$\\#\\SDB$, and $d$ $=$ $\\#\\I$. \nComputing the pseudo-projected database $\\mathcal{PSDB}_i$ can be\ndone in $O(m \\times \\ell)$: for \neach sequence $(sid,s)$ of $\\SDB$, checking if $\\sigma$ occurs in $s$\nis $O(\\ell)$ and there are $m$ sequences. The total complexity of\nfunction \\codex{GetFreqItems} is $O(m\\times(\\ell + d))$.  \nLines (\\ref{proj8}-\\ref{proj10}) can be achieved in $O(\\ell\\times d)$.\nSo, the whole complexity is $O(m \\times \\ell + m\\times(\\ell + d) +\n\\ell\\times d)$ $=$ $O(m\\times\\ell + m\\times d + \\ell\\times d)$. \nThe space complexity of the filtering algorithm lies in the storage of\nthe $\\mathcal{PSDB}$ internal data structure. In the worst case, we have to\nstore $\\ell$ pseudo-projected databases. Since each pseudo-projected database\nrequires $O(m)$, the worst case space complexity is $O(m\\times \\ell)$.  \n$\\Box$\n\n\n\\subsection{Encoding of SPM Constraints}\n\\label{pattern-encoding}\nUsual SPM constraints (see Section~\\ref{sec:local}) can be reformulated in a straightforward way.\nLet $P$ be the unknown pattern.\n \n\\noindent- {\\it Minimum size constraint}:\n$size(P,\\ell_{min})  \\equiv\n\\bigwedge_{i=1}^{i=\\ell_{min}} (P_i \\neq \\square)$\n\n\\noindent- {\\it Item constraint}: let $V$ be a subset of items, $l$ and $u$\ntwo integers  s.t. $0 \\leq l\\leq u \\leq \\ell$.  \n\\hspace*{-.07cm}$item(P,V) \\equiv \\bigwedge_{t \\in V} \\mbox{\\tt Among}(P,\\{t\\},l,u)$\nenforces that items of $V$ should occur at least $l$ times and at most $u$ times in $P$.\nTo forbid items of $V$ to occur in $P$, $l$ and $u$ must be set to $0$. \n\n\\noindent \n- {\\it Regular expression constraint}:\nlet $A_{reg}$ be the deterministic finite automaton encoding the regular\nexpression $exp$.\n$reg(P,exp) \\equiv {\\tt Regular}(P, A_{reg})$.\n\n\n\\subsection{Experimental protocol} \nThe implementation of our approach was carried out in the {\\tt Gecode}\nsolver\\footnote{\\url{http:\/\/www.gecode.org}}. \nAll experiments were conducted on a machine with a processor Intel X5670 and 24 GB of memory. \nA time limit of 1 hour has been used. \nFor each dataset, we varied the $minsup$ threshold until the\nmethods are not able to complete the extraction of all patterns\nwithin the time limit. \n$\\ell$ was set to the length of the longest\nsequence of $\\SDB$. The implementation and the datasets used in our\nexperiments are available\nonline\\footnote{\\url{https:\/\/sites.google.com\/site\/prefixprojection4cp\/}}. \nWe compare our approach (indicated by \\texttt{PP}\\xspace) with:\n\\begin{enumerate} \n\\item two CP encodings~\\cite{NegrevergneCPIAOR15}, the most efficient CP\n  methods for SPM: \\texttt{global-p.f}\\xspace and \n  \\texttt{decomposed-p.f}\\xspace; \n\\item state-of-the-art methods for SPM : \\texttt{PrefixSpan}\\xspace and \\texttt{cSpade}\\xspace; \n\\item \\sma~\\cite{Bonchi:2008} for SPM under regular expressions. \n\\end{enumerate}\n\nWe used the author's \\texttt{cSpade}\\xspace\nimplementation~\\footnote{\\url{http:\/\/www.cs.rpi.edu\/~zaki\/www-new\/pmwiki.php\/Software\/}} \nfor SPM, the publicly available implementations of \\texttt{PrefixSpan}\\xspace by\nY. Tabei~\\footnote{\\url{https:\/\/code.google.com\/p\/prefixspan\/}} and \nthe \\sma\nimplementation~\\footnote{\\url{http:\/\/www-kdd.isti.cnr.it\/SMA\/}} for\nSPM under regular expressions. The\nimplementation~\\footnote{\\url{https:\/\/dtai.cs.kuleuven.be\/CP4IM\/cpsm\/}}\nof the two CP encodings was carried out in the {\\tt Gecode} solver. \nAll methods have been executed on the same machine.\n\n\\begin{figure*}[t]\t\n{\\footnotesize\n\\begin{tabular}{ccc}\nBIBLE & Kosarak & PubMed \\\\\n\t\\includegraphics[width=3.8cm, height=2.8cm]{BIBLECPSM.pdf}\n&   \n\\includegraphics[width=3.8cm, height=2.8cm]{KosarakCPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{PUBMEDCPSM.pdf}\n\\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{numberBIBLE.pdf}\n&   \n\\includegraphics[width=3.8cm, height=2.8cm]{numberKOSARAK.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberPUBMED.pdf}\n\\\\\nFIFA & Leviathan & Protein \\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{FIFACPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{LeviathanCPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{PROTEINCPSM.pdf}\n\\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{numberFIFA.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberLEVIATHAN.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberPROTEIN.pdf}\n\\end{tabular}}\n\\vspace*{-.35cm}\n \n\\caption{\\small \\label{fig:FreqCPSM} Comparing \\texttt{PP}\\xspace with \\texttt{global-p.f}\\xspace\n  for SPM on real-life datasets: CPU times (top) and number of patterns (bottom).} \n\\end{figure*}\n\n\n\\subsection{Comparing with CP Methods for SPM}\nFirst we compare \\texttt{PP}\\xspace with the two CP encodings \n  \\texttt{global-p.f}\\xspace and \\texttt{decomposed-p.f}\\xspace (see Section\n  \\ref{CP4SPM}). \nFig.~\\ref{fig:FreqCPSM} shows the number of extracted sequential patterns and the CPU times\nto extract them (in logscale for BIBLE, Kosarak and PubMed) for the\nthree methods. \n\nFirst, as expected, the lower $minsup$ is, the larger the number of\nextracted sequential patterns. \nSecond, when comparing the CPU times, \\texttt{decomposed-p.f}\\xspace is the least performer\n  method. On all the datasets, it fails to complete the extraction within the time \n  limit for all values of $minsup$ we considered. \nThird, \\texttt{PP}\\xspace largely dominates \\texttt{global-p.f}\\xspace on all the datasets: \n\\texttt{PP}\\xspace is more than an order of magnitude faster than \\texttt{global-p.f}\\xspace. \nThe gains in terms of CPU times are greatly amplified for low values\nof $minsup$. On BIBLE (resp. PubMed), the speed-up is\n$84.4$ (resp. $33.5$) for $minsup$ equal to $1\\%$. \nAnother important observation that can be made is that, on most of the\ndatasets (except BIBLE and Kosarak), \\texttt{global-p.f}\\xspace is not able to mine for\npatterns at very low frequency within the time limit. \nFor example on FIFA, \\texttt{PP}\\xspace is able to\ncomplete the extraction for values of $minsup$ up to $6\\%$ in $1,457$ \nseconds, while \\texttt{global-p.f}\\xspace fails to complete the extraction for $minsup$\nless than $10\\%$. The same trend is also conformed on Leviathan, where\n\\texttt{global-p.f}\\xspace is not able to mine for patterns at $1\\%$ minimum frequency.  \n\n\nTo complement the results given by  Fig.~\\ref{fig:FreqCPSM},\nTable~\\ref{table:SolverStat} reports for different datasets and different\nvalues of $minsup$, the number of calls to the propagate routine of\n{\\tt Gecode} (column 5), and the number of nodes of the search tree (column\n6). \nFirst, \\texttt{PP}\\xspace explores less nodes than \\texttt{global-p.f}\\xspace. But, the difference is\nnot huge (gains of 45\\%  and 33\\% on FIFA and BIBLE\nrespectively). \nSecond, our approach is very\neffective in terms of number of propagations. For \\texttt{PP}\\xspace, the number of \npropagations remains small (in thousands for small values of $minsup$)\ncompared to  \\texttt{global-p.f}\\xspace (in millions). \nThis is due to \nthe huge number of reified constraints used in \\texttt{global-p.f}\\xspace to encode the\nsubsequence relation. On the contrary, our \\textsc{Prefix-Projection}\\xspace global constraint\ndoes not require any reified constraints nor any extra variables to\nencode the subsequence relation.\n\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$minsup$ (\\%)} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{FIFA} &20 & 938 & {\\bf 8.16} & 129.54 & {\\bf 1884} & 11649290 & {\\bf 1025} & 1873 \\\\\n&18 & 1743 &{\\bf 13.39} & 222.68 & {\\bf 3502} & 19736442  & {\\bf 1922} & 3486 \\\\\n&16 & 3578 & {\\bf 24.39} & 396.11 &{\\bf 7181} & 35942314 & {\\bf 3923} & 7151\\\\\n&14 & 7313 & {\\bf 44.08} & 704 & {\\bf 14691} & 65522076 & {\\bf 8042} & 14616\\\\\n&12 & 16323 & {\\bf 86.46} & 1271.84 & {\\bf 32820} & 126187396 & {\\bf 18108} & 32604 \\\\\n&10 & 40642 & {\\bf 185.88} & 2761.47 & {\\bf 81767 } & 266635050  & {\\bf 45452} & 81181  \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{BIBLE} &10 & 174 &{\\bf 1.98}  & 105.01 & {\\bf 363}  & 4189140& {\\bf 235} & 348 \\\\\n&8 & 274 & {\\bf 2.47}  & 153.61 & {\\bf 575}  & 5637671 & {\\bf 362} & 548 \\\\\n&6 & 508 & {\\bf 3.45}  & 270.49 & {\\bf 1065} & 8592858 & {\\bf 669} & 1016 \\\\\n&4 & 1185 & {\\bf 5.7}  & 552.62 & {\\bf 2482} & 15379396 & {\\bf 1575}  & 2371 \\\\\n&2 & 5311 & {\\bf 15.05}  & 1470.45 & {\\bf 11104}  & 39797508  &  {\\bf 7048} & 10605  \\\\ \n&1 & 23340 & {\\bf 41.4} & 3494.27 & {\\bf 49057}  & 98676120 & {\\bf 31283} & 46557 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{PubMed} &5 &  2312 & {\\bf 8.26} &  253.16 & {\\bf 4736}   & 15521327  & {\\bf 2833}  &  4619 \\\\ \n&4 & 3625 & {\\bf 11.17}  & 340.24 &{\\bf 7413} & 20643992 & {\\bf 4428}  & 7242 \\\\\n&3 & 6336 & {\\bf 16.51} & 536.96 &{\\bf 12988} &29940327  & {\\bf 7757} & 12643 \\\\\n&2 & 13998 & {\\bf 28.91} & 955.54 & {\\bf 28680} &50353208  & {\\bf 17145} & 27910  \\\\\n&1 & 53818 & {\\bf 77.01} & 2581.51 & {\\bf 110133} &  124197857 &  {\\bf 65587} &  107051 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Protein} &99.99 & 127 &{\\bf 165.31} & 219.69 & {\\bf 264} & 26731250 & {\\bf 172} & 221 \\\\\n&99.988 & 216 & {\\bf 262.12} & 411.83 & {\\bf 451} & 44575117& {\\bf 293} & 390\\\\\n&99.986 & 384 & {\\bf 467.9}6 & 909.47& {\\bf 805} &80859312 & {\\bf 514} & 679\\\\\n&99.984 & 631 & {\\bf 753.3} & 1443.92 & {\\bf 1322} & 132238827 & {\\bf 845} & 1119\\\\\n&99.982 & 964& {\\bf 1078.73} & 2615 & {\\bf 2014} & 201616651 &  {\\bf 1284} & 1749\\\\ \n&99.98 & 2143 & {\\bf 2315.65} & {\\bf $-$} & {\\bf 4485}  & {\\bf $-$} & {\\bf 2890} & {\\bf $-$} \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Kosarak} &1 & 384 &{\\bf 2.59} & 137.95 & {\\bf 793} & 8741452 & {\\bf 482} & 769\\\\\n&0.5 & 1638 & {\\bf 7.42} & 491.11 & {\\bf 3350} & 26604840 & {\\bf 2087} & 3271\\\\\n&0.3 & 4943& {\\bf 19.25} & 1111.16& {\\bf 10103} & 56854431& {\\bf 6407} &  9836\\\\\n&0.28 & 6015 & {\\bf 22.83} & 1266.39 & {\\bf 12308} & 64003092 & {\\bf 7831} & 11954\\\\\n&0.24 & 9534 & {\\bf 36.54} & 1635.38 & {\\bf 19552} & 81485031 &  {\\bf 12667} & 18966 \\\\\n&0.2  & 15010& {\\bf 57.6} & 2428.23 & {\\bf 30893} & 111655799 &  {\\bf 20055} & 29713 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Leviathan} &10 & 651 &{\\bf 1.78} & 12.56 & {\\bf 1366} &  2142870& {\\bf 849} & 1301\\\\\n&8 & 1133 & {\\bf 2.57} & 19.44 & {\\bf 2379} &3169615  & {\\bf 1487} & 2261\\\\\n&6 & 2300 & {\\bf 4.27} &32.85 & {\\bf 4824} &5212113 & {\\bf 3008} &4575 \\\\\n&4 & 6286 & {\\bf 9.08} &  66.31& {\\bf 13197} &10569654  & {\\bf 8227} &12500  \\\\\n&2 & 33387& {\\bf 32.27} & 190.45& {\\bf 70016} &  33832141&  {\\bf 43588} &66116 \\\\ \n&1 & 167189& {\\bf 121.89} & $-$ & {\\bf 350310} & $-$ &  {\\bf 217904} & $-$\\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\\texttt{PP}\\xspace vs. \\texttt{global-p.f}\\xspace.} \n\\label{table:SolverStat}\n\\end{table*}\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{tabular}{ccc}\nBIBLE & Kosarak & PubMed   \\\\\n\\includegraphics[width=4cm, height=3.0cm]{BIBLEC.pdf\n&   \n\\includegraphics[width=4cm, height=3.0cm]{KosarakC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{PUBMEDC.pdf\n\\\\\nFIFA & Leviathan & Protein \\\\\n\\includegraphics[width=4cm, height=3.0cm]{FIFAC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{LeviathanC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{PROTEINC.pdf}\n\\vspace*{-.35cm}\n\\end{tabular}}\n\n\\caption{\\small \\label{fig:FreqCLASSIC} Comparing \\textsc{Prefix-Projection}\\xspace with\n  state-of-the-art algorithms for SPM.} \n\\end{figure*}\n\n\\subsection{Comparing with ad hoc Methods for SPM}\nOur second experiment compares \\texttt{PP}\\xspace with state-of-the-art methods  \nfor SPM. Fig.~\\ref{fig:FreqCLASSIC} shows the CPU times of the three \nmethods.\nFirst, \\texttt{cSpade}\\xspace obtains the best\nperformance on all datasets (except on Protein).  \nHowever, \\texttt{PP}\\xspace exhibits a similar behavior as \\texttt{cSpade}\\xspace, but it is less\nfaster (not counting the highest values of $minsup$).  \nThe behavior of \\texttt{cSpade}\\xspace on Protein is due to the\n  vertical representation format that is not appropriated in the\n  case of databases having large sequences and small number of\n  distinct items, thus degrading the performance of the mining\n  process.\nSecond, \\texttt{PP}\\xspace which also uses the concept of projected databases,\nclearly outperforms \\texttt{PrefixSpan}\\xspace on all datasets. \nThis is due to\nour filtering algorithm combined together with incremental data\nstructures to manage the projected databases.  \nOn FIFA, \\texttt{PrefixSpan}\\xspace is not\nable to complete the extraction for $minsup$ less than $12\\%$, while\nour approach remains feasible until $6\\%$ within the time limit. \nOn Protein, \\texttt{PrefixSpan}\\xspace fails to complete the\n  extraction for all values of $minsup$ we considered. \nThese results clearly demonstrate that our approach competes well with\nstate-of-the-art methods for SPM on large datasets and achieves scalability \nwhile it is a major issue of existing CP approaches. \n\n\\subsection{SPM under size and item constraints}\nOur third experiment aims at assessing the interest of pushing\nsimultaneously different types of constraints. We impose on the PubMed\ndataset usual constraints such as {\\it the minimum frequency} and the\n{\\it minimum size} constraints and other useful constraints expressing\nsome linguistic knowledge such as {\\it the item\n  constraint}. The goal is to retain sequential patterns which convey\nlinguistic regularities (e.g., gene - rare disease relationships)~\\cite{BCCC2012cbms}.  \n{\\it The size constraint} allows to\nremove patterns that are too small w.r.t. the\nnumber of items (number of words) to be relevant patterns. We tested\nthis constraint with $\\ell_{min}$ set to 3.  \n{\\it The item constraint} imposes that the extracted patterns must\ncontain the item GENE and the item DISEASE. \nAs no ad hoc method exists for this combination of \nconstraints, we only compare \\texttt{PP}\\xspace with\n\\texttt{global-p.f}\\xspace. \nFig.~\\ref{fig:const} shows the CPU times and the number of\nsequential patterns extracted \nwith and without constraints. \nFirst, pushing simultaneously the two constraints enables to reduce\nsignificantly the number of patterns. Moreover, the CPU times for \\texttt{PP}\\xspace decrease \nslightly whereas for \\texttt{global-p.f}\\xspace (with and without\nconstraints), they are almost the same. \nThis\nis probably due to the weak communication between\nthe $m$ \\texttt{exists-embedding} reified global constraints and the\ntwo constraints. This reduces significantly the quality of the\nwhole filtering. \nSecond (see Table~\\ref{table:const:stat}), when considering the\ntwo constraints, \\texttt{PP}\\xspace clearly dominates \\texttt{global-p.f}\\xspace (speed-up value up to\n$51.5$). Moreover, the number of propagations performed by \\texttt{PP}\\xspace remains \nvery small as compared to \\texttt{global-p.f}\\xspace. \nFig.~\\ref{fig:const:prop} compares the two methods under \nthe minimum size constraint for different values of \n$\\ell_{min}$, with $minsup$ fixed to\n$1\\%$. Table~\\ref{table:const:stat} compares the two methods in terms\nof numbers of propagations (column $5$) and number of nodes of the\nsearch tree (column $6$). \nOnce again, \\texttt{PP}\\xspace is always the most performer method (speed-up\nvalue up to $53.1$). These results also confirm what we observed previously,\nnamely the weak communication between reified global constraints \nand constraints imposed on patterns (i.e., size and item constraints). \n\n\\begin{figure*}[t]\n\\centering\n\\subfloat[\\# of patterns\\label{fig:const:patterns}]{\n\\includegraphics[width=4cm, height=2.6cm]{numberPP-CPSM-const.pdf}\n}\n\\subfloat[CPU times (logscale) \\label{fig:const:time}]{\n\\includegraphics[width=4cm, height=2.8cm]{PP-CPSM-const.pdf}\n}\n\\subfloat[Minimum size constraint\\label{fig:const:prop}]{\n\\includegraphics[width=4cm, height=2.6cm]{PP-CPSM-size.pdf}\n}\n\\vspace*{-.35cm} \n\\caption{\\small Comparing \\texttt{PP}\\xspace with \\texttt{global-p.f}\\xspace under minimum size and item constraints on PubMed.} \\label{fig:const}\n\\end{figure*}\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$minsup$ (\\%)} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{PubMed} & 5 & 279 & {\\bf 6.76} & 252.36 & {\\bf 7878} & 12234292  & {\\bf 2285} & 4619  \\\\ \n&4 & 445 & {\\bf 8.81}  & 339.09 & {\\bf 12091} & 16475953 & {\\bf 3618}  & 7242 \\\\\n&3 & 799 & {\\bf 12.35} & 535.32 & {\\bf 20268} & 24380096 & {\\bf 6271}   & 12643 \\\\\n&2& 1837 & {\\bf 20.41} & 953.32 & {\\bf 43088}   &  42055022 & {\\bf 13888}  & 27910  \\\\ \n&1 & 7187 & {\\bf 49.98} & 2574.42 & {\\bf 157899}  &  107978568 & {\\bf 52508}  &  107051 \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\\small {\\tt PP} vs. \\texttt{global-p.f}\\xspace under minimum size and item constraints.} \n\\label{table:const:stat}\n\\end{table*}\n\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$\\ell_{min}$} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{PubMed} & 8 & 12 & {\\bf 48.52} & 2577.09 & {\\bf 55523} & 105343528  & {\\bf 50264} &  107051\\\\ \n& 6 & 3596 & {\\bf 50.91} & 2576.9 & {\\bf 59144} & 106272419  & {\\bf 50486} & 107051 \\\\ \n& 4 & 40669 & {\\bf 70.61} & 2579.3 & {\\bf 96871} & 117781215  & {\\bf 59194} &  107051 \\\\ \n& 2 & 53486 & {\\bf 76.64} & 2580.41  & {\\bf 109801} &  123913176 & {\\bf 65334} & 107051 \\\\ \n& 1 & 53818 & {\\bf 78.49} & 2579.85  & {\\bf 110133} &  117208559 & {\\bf 65587} &  107051\\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\\small {\\tt PP} vs. \\texttt{global-p.f}\\xspace under minimum size constraint.} \n\\label{table:const:stat}\n\\end{table*}\n\n\\subsection{SPM under regular constraints}\nOur last experiment compares \\prefixCPREG against two variants of \\sma:\n\\smap (\\sma one pass) and\n\\texttt{SMA-FC}\\xspace (\\sma Full Check). \nTwo datasets are considered from~\\cite{Bonchi:2008}: one synthetic\ndataset ({data-200k}), and one real-life dataset (Protein). \nFor {data-200k}, we used two RE: \n\\begin{itemize}\n\\item $\\mathtt{RE}10 \\equiv A^*B(B|C)D^*EF^*(G|H)I^*$, \n\\item $\\mathtt{RE}14 \\equiv A^*(Q|BS^*(B|C))D^*E(I|S)^*(F|H)G^*R$. \n\\end{itemize}\n\nFor {Protein}, we used $\\mathtt{RE}2 \\equiv (S|T)$ $.$ $(R|K)$\nrepresenting {\\it Protein kinase C phosphorylation} \n(where $.$ represents any symbol).  \nFig.~\\ref{fig:reg} reports CPU-times comparison. \nOn the synthetic dataset, our approach is very effective. \nFor $\\mathtt{RE}14$, our\nmethod is more than an order of magnitude faster than \\sma. \nOn Protein, the gap between the $3$ methods\nshrinks, but our method remains effective. For the particular\ncase of $\\mathtt{RE}2$, the {\\tt Regular} constraint can  be substituted\nby restricting the domain of the first and third variables to\n$\\{S,T\\}$ and $\\{R,K\\}$ respectively (denoted as \\texttt{PP-SRE}\\xspace), thus\nimproving performances. \n\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{tabular}{ccc}\ndata-200k (RE10) & data-200k (RE14) & Protein (RE2) \\\\  \n\\includegraphics[width=4cm, height=3cm]{REG-data200k-RE10-col.pdf}\n&   \n\\includegraphics[width=4cm, height=3cm]{REG-data200k-RE14-col.pdf}\n&\n\\includegraphics[width=4cm, height=3cm]{REG-protein-col.pdf}\n\t\\end{tabular}}\n\\vspace*{-.35cm} \n\\caption{\\small \\label{fig:reg} Comparing \\textsc{Prefix-Projection}\\xspace with \\sma for SPM under RE constraint.}\n\\end{figure*}\n\n\n\n\\section{Introduction}\n\\label{section:introduction}\n\\input{1-introduction.tex}\n\n\\section{Preliminaries}\n\\label{section:preliminaries}\n\\input{2-context.tex}\n\n\\vspace*{-.15cm}\n\\section{Related works}\n\\label{PB-related-works}\n\\input{3-relatedworks.tex}\n\n\\section{\\textsc{Prefix-Projection}\\xspace Global Constraint}\n\\label{sec:model}\n\\input{4-projection.tex}\n\n\n\\section{Experimental Evaluation}\n\\label{section:experimentations}\n\\input{6-experiments.tex}\n\n\\section{Conclusion}\n\\label{section:conclusion}\n\\input{7-conclusion.tex}\n\n\\newpage\n\\bibliographystyle{splncs03}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum Hamiltonians modelling condensed matter typically exhibit a lattice structure. In the idealized setting one uses operators which are periodic { with respect to  } the lattice, while more realistic models, taking into account disorder present in the model, consist in random operators which are homogeneous and ergodic { with respect to  } lattice translations. In the framework of quantum mechanics the key  to understanding properties of the underlying physical system is the spectrum of the operator.  It is a fundamental fact in the theory of random operators that the spectrum of an ergodic or metrically-transitive ensemble of operators is almost surely non-random, in the sense that there exists a fixed set $\\Sigma \\subset \\RR$ such that for almost all elements in the ensemble, the spectrum of the operator coincides with $\\Sigma$.\nThis holds even in an abstract Hilbert space setting, cf. e.g. \\cite{KirschM-82a}.\n\n\nThe structure or shape of the spectrum as a subset of the real axis and its measure theoretic features are intimately connected to conductance and transport properties of the condensed matter.\nWith the emergence of disorder the spectrum of as a set grows. In order to understand this expansion in a quantitative way, one introduces a scalar parameter coupling the random part of the Hamiltonian to the periodic background operator.  In the simplest case the coupling is linear.  The question of interest in the present paper is: At what rate does the spectrum expand, as the scalar coupling increases? This is the first step in the analysis of more delicate spectral properties.\nIndeed, it is precisely the new portions of the spectrum generated by the random perturbation of the Hamiltonian where interesting phenomena (like a transition from point to continuous spectrum) are expected to occur. We consider here only the expansion of spectrum at its lowest edge, although it seems that a refinement of our methods together with substantially more technical effort would apply to general spectral edges as well.\n\n\nThe \\emph{motivation for the present work} is the following:\nThe most commonly encountered situation for weak-disorder random Schr\\\"odinger operators\nis that the spectrum expands linearly with the disorder --- provided that sign conspiracy does not prevent expansion altogether.\n(If $-\\Delta$ is perturbed by a non-negative potential, there is obviously no expansion of the spectrum.)\nHowever, there are examples where the expansion is quadratic in the disorder parameter. So it is natural to ask whether even\n\\emph{slower growth} (cubic, quartic, \\ldots) \\emph{is possible to occur?}\nIn the most important case, growth can occur only either with a linear or a quadratic rate,\nas we discuss in Corollary \\ref{c:1}  and Remark \\ref{r:1} below.\n\n\n\nMore generally, we derive in the present paper\nestimates on the expansion of the spectrum which are in line  with earlier research,\nin particular on the weak disorder regime, {e.~g.}\n\\cite{aizenman1994localization}, \\cite{wang2001},\n\\cite{klopp2002a}, \\cite{klopp2002b}, \\cite{elgart2009lifshitz}, \\cite{elgartetal2011locnonmono}, \\cite{caoelgart2012}, \\cite{hoeckerescuti2013}, \\cite{hoeckerescuti2014},\n\\cite{borisovveselic2011}, \\cite{borisovveselic2013}. Let us give some more details:\nIn \\cite{aizenman1994localization} the fractional moment method is developed to show localization for the Anderson model in the weak disorder regime,\n\\cite{wang2001} treats the Anderson model as well and gives an estimate on the size of the energy interval where localization occurs as a function of the disorder parameter.\nThis bound has been refined in \\cite{klopp2002a} and \\cite{elgart2009lifshitz}, while a similar result for random operator on $\\RR^d$ has been derived in\n\\cite{klopp2002b}.\nIn the later paper the single site potential is allowed to change sign, but needs to have a non-vanishing average.\nIn the context of the present paper, this means the spectral edge is shifted linearly as a function of the disorder parameter.\nDiscrete alloy type models on $\\ZZ^d$ have been analyzed in \\cite{elgartetal2011locnonmono},\nand for $\\ZZ^3$ an analogous result to \\cite{klopp2002b} has been derived in \\cite{caoelgart2012}. It covers also the case\nwhen the average of the single site potential is zero, which requires a more detailed analysis.\nIn particular,  this means the spectral edge is shifted quadratically as a function of the disorder parameter.\nA particular regime where one can prove Anderson localization is the Lifshitz tail region near the spectral bottom.\nUpper and lower bounds on the size of the region (again as a function of the disorder parameter) have been given in\n\\cite{hoeckerescuti2013} and \\cite{hoeckerescuti2014}. The results mentioned so far concern models where a random potential\nis coupled linearly to a kinetic energy term. In contrast to this \\cite{borisovveselic2011} and \\cite{borisovveselic2013} analyse\nlow lying eigenvalues of (long, finite segments of) randomly shifted and curved waveguides, respectively.\nIn this case the random perturbation term depends in a nonlinear way on the random variables and\nthe disorder parameter and consist of (lower order) differential operators.\n\nThe present paper\nprovides a fundamental analysis about the location of the spectrum which is a prerequisite for identifying the energy  region\nof Anderson localization.\nFor a broader discussion of the physical intuition and the relevance of our result to the general understanding of spectral properties of random Hamiltonians\nwe refer to the discussion in our previous work \\cite{BorisovHEV}.\nThere we have carried out an analogous analysis for discrete Hamiltonians, i.e. matrix operators over $\\ell^2(\\ZZ^d)$. In the discrete setting one has less technical questions to take care of, for instance properly defined domains or compactness properties, either because certain operators are automatically bounded, or because auxiliary Hilbert spaces are finite dimensional.\nNevertheless, our approach carries over to a very general class of operators in continuum space. In the present paper we chose not to present the most general model class (which we may do in some later manuscript) but to restrict ourselves to the case that the unperturbed periodic part of the Hamiltonian is a differential operator, and the perturbations satisfy certain relative boundedness conditions. This framework has the advantage that it on one hand covers a variety of physically relevant cases, but on the other hand is specific enough to avoid a long list of abstract hypotheses.\n\n\nWe close the section\nhighlighting  the general scope, the flexible versatility, and the technical advancements of our approach.\nIn comparison to mentioned previous work on spectral properties of random Schr\\\"odinger type operators in the weak-disorder regime\nthe present paper has the following new features\n\\begin{itemize}\n \\item The results are formulated for very general types of random operators (actually quadratic forms).\n\n \\item {The results require only weak assumptions on the random variables entering the model. In particular, the range of values need not be of fixed sign, nor an interval.}\n\n \\item {No monotonicity condition for the random perturbations is assumed.}\n\n \\item {We prove both upper and lower bounds for the bottom of the perturbed spectrum.}\n\\end{itemize}\nThis is motivated by the fact that the mechanism which determines the expansion of the spectrum is\nof quite universal nature. So it makes more sense to establish it once and for all models of practical importance,\nrather than to repeat the argument in each setting separately.\n\n\n\n\\section{Model and main results}\n\n\nLet $x=(x_1,\\ldots,x_d)$ be Cartesian coordinates in $\\mathds{R}^d$, $d\\geqslant d_1\\geqslant 1$, $e_1,\\ldots,e_{d_1}$ be linearly independent vectors in $\\mathds{R}^d$, and $\\G$ be the lattice $\\{x\\in\\mathds{R}^d:\\, x=z_1 e_1 +\\ldots + z_{d_1}e_{d_1},\\,z_i\\in \\mathds{Z}\\}$. By $\\Pi$ we denote an infinite domain in $\\mathds{R}^d$ which is invariant  with respect to   shifts along $\\G$, namely, for each $x\\in\\Pi$, $k\\in\\G$ we have $x-k\\in\\Pi$.\nLet $\\square$ be a periodicity cell of $\\Pi$, i.e., a minimal domain such that $\\Pi$ is the interior of $\\overline{\\bigcup\\limits_{k\\in\\G} \\square_k}$, $\\square_k:=\\{x\\in\\mathds{R}^d:\\, x-k\\in \\square \\}$.\n\nLet $m\\geqslant 1$ be a given natural number,\n$A_{\\a\\b}=A_{\\a\\b}(x)$, $\\a,\\b\\in\\mathds{Z}_+^d$, $|\\a|,|\\b|\\leqslant m$, be functions defined on $\\Pi$ satisfying the conditions:\n\\begin{equation}\\label{2.1}\n\\begin{aligned}\n&A_{\\a\\b}\\ \\text{are $\\square$-periodic},\\quad A_{\\a\\b}\\in C^{|\\a|}(\\overline{\\Pi}),\n\\quad \\overline{A_{\\b\\a}}=A_{\\a\\b}\n\\\\\n&\\sum\\limits_{\\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|=|\\b|=m}} A_{\\a\\b}(x)\\xi^{\\a+\\b}\n\\geqslant c_\n|\\xi|^{2m},\\quad x\\in\\overline{\\Pi},\\quad \\xi\\in\\mathds{R}^d,\n\\end{aligned}\n\\end{equation}\nwhere for $\\xi=(\\xi_1,\\ldots,\\xi_d)$, $\\a+\\b=(\\a_1+\\b_1,\\ldots,\\a_2+\\b_2)$ we denote $\\xi^{\\a+\\b}:=\\xi_1^{\\a_1+\\b_1}\\cdot\\ldots\\cdot\\xi_d^{\\a_d+\\b_d}$,\nand $c_0$ is a fixed positive constant independent of $x$ and $\\xi$.\nThe boundary of the domain $\\Pi$ is assumed to be $C^{2m}$-smooth.\nIn $L_2(\\Pi)$ we consider the operator\n\\begin{equation}\\label{2.2}\n\\Op_0:=  \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}\\p^\\a A_{\\a\\b} \\p^\\b,\\quad \\p^\\a:=\\frac{\\p^\\a\\,}{\\p x^\\a}\n\\end{equation}\nsubject to the Dirichlet condition\n\\begin{equation}\\label{2.3}\nu=0,\\quad \\frac{\\p^j u}{\\p\\nu^j}=0,\\quad \\text{on}\\quad \\p\\Pi,\\quad j=1,\\ldots,m-1,\n\\end{equation}\nwhere $\\nu$ is the outward normal to $\\p\\Pi$.\nThanks to  conditions (\\ref{2.1}), the operator $\\Op_0$ is self-adjoint, elliptic, and lower  semi-bounded\non the domain\n\\begin{equation*}\n\\Dom(\\Op_0):=\\{u\\in W_2^{2m}(\\Pi):\\, \\text{boundary conditions (\\ref{2.3}) are satisfied}\\}.\n\\end{equation*}\nThe associated sesquilinear form is\n\\begin{equation}\\label{2.4}\n\\fm_0(u,v)= \\sum\\limits_{ \\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (A_{\\a\\b} \\p^\\b u , \\p^\\a v )_{L_2(\\Pi)} \\quad\\text{in}\\quad L_2(\\Pi)\n\\end{equation}\non the domain\n\\begin{equation*}\n\\Dom(\\fm_0):=\\{u\\in W_2^{m}(\\Pi):\\, \\text{boundary conditions (\\ref{2.3}) are satisfied}\\}.\n\\end{equation*}\nThe above stated facts about $\\Op_0$ can be proven in the same way as for second order differential operators,\none should just  employ appropriate smoothness improving theorems, see \\cite[Ch. III, Sect. 6, Lm. 6.3]{Ber}.\n\n\nLet  $\\omega_k$, $k\\in\\G$, be a sequence  of bounded, non-trivial, independent, identically distributed random variables with distribution measure $\\mu$. We assume that\n\\begin{equation*}\n\\{s_-,s_+\\}\\in \\supp\\mu \\subseteq [s_-,s_+],\n\\end{equation*}\nwhere the\nnumbers $s_\\pm$ satisfy one of the following alternatives:\n\\begin{equation}\\label{2.7}\ns_-<0<s_+\n\\end{equation}\nor\n\\begin{equation}\\label{2.8}\n0\\leqslant s_-<s_+.\n\\end{equation}\nWithout loss of generality we suppose that each random variable $\\omega_k$ is defined on the probability space $(\\supp \\mu,\\mathfrak{B},\\mu)$, where $\\mathfrak{B}$ is the Borel $\\si$-algebra on $\\supp \\mu\\subset [s_-,s_+]$,\nand $\\omega_k$ are just the identity mappings on $\\supp \\mu$.\nThus the random field  $(\\omega_k)_{k\\in\\G}$\nis an element of the product probability space $(\\Om, \\bigotimes_\\G \\mathfrak{B}, \\PP)$ with\nmeasure $\\PP:=\\bigotimes_{k\\in\\G} \\mu$ and configuration space $\\Om:=\\times_{k\\in\\G}\\supp \\mu$.\n\nWe let $\\g_\\Pi:=\\p\\square\\cap\\p\\Pi$, $\\g_l:=\\p\\square\\setminus\\p\\Pi$.\nBy $\\Ho^{2m}(\\square,\\g_\\Pi)$ we denote the subspace of $W_2^{2m}(\\square)$ consisting of functions satisfying the boundary conditions\n\\begin{equation}\\label{2.5}\nu=0,\\quad \\frac{\\p^j u}{\\p\\nu^j}=0\\quad \\text{on}\\quad \\g_\\Pi,\\quad j=1,\\ldots,m-1.\n\\end{equation}\nLet $\\cL(t)$, $t\\in[-t_0,t_0]$, $t_0>0$, be a family of operators from $\\Ho^{2m}(\\square,\\g_\\Pi)$ into $L_2(\\square)$ defined as\n\\begin{equation}\\label{2.6}\n\\cL(t)=t \\cL_1 + t^2 \\cL_2 + t^3 \\cL_3(t),\n\\end{equation}\nwhere $\\cL_i$ are bounded symmetric operators from $\\Ho^{2m}(\\square,\\g_\\Pi)$ into $L_2(\\square)$;\nmoreover,  the operator $\\cL_3(t)$ is assumed to be bounded uniformly in $t$.\nWe also suppose that the map\n$$\nt \\mapsto \\left(\\cL_3(t): \\Ho^{2m}(\\square, \\g_\\Pi)\\to L_2(\\square)\\right)\n$$\nis continuous in $t\\in[-t_0,t_0]$.\n\nBy $\\cS(k)$, $k\\in\\G$, we denote the shift operator: $(\\cS(k)u)(x):=u(x-k)$.\nThe main operator of our study is\n\\begin{equation}\\label{2.8a}\n\\Op_\\e(\\om):=\\Op_0 + \\cL_\\e(\\om),\n\\quad \\om:=(\\omega_k)_{k\\in\\G},\\quad \\cL_\\e(\\om):=\\sum\\limits_{k\\in\\G} \\cS(k) \\cL(\\e\\omega_k) \\cS(-k),\n\\end{equation}\nin $L_2(\\Pi)$ on the domain $\\Dom(\\Op_\\e(\\om)):=\\Dom(\\Op_0)$. Here $\\e$ is a small positive parameter.\n\n\nLet us clarify the action of the operator $\\cL_\\e(\\om)$. Given $u\\in\\Dom(\\Op_0)$, it is clear that the restriction of $u$ on $\\square_k$\nbelongs to $\\Ho^{2m}(\\square_k,\\g_\\Pi^k)$, $\\g_\\Pi^k:=\\p\\square_k\\cap\\p\\Pi$. Then $(\\cS(-k)u)(x)=u(x+k)$ is an element of $\\Ho^{2m}(\\square,\\g_\\Pi)$ and the action of $\\cL(\\e\\omega_k)$ on $\\cS(-k)u$ is well-defined as an element of $L_2(\\square)$.\nBy applying $\\cS(k)$ to the result of the action, we just shift the function $\\cL(\\e\\omega_k)\\cS(-k)u$ to  the cell $\\square_k$.\nIn this way each operator $\\cS(k) \\cL(\\e\\omega_k) \\cS(-k)$ acts on $\\Ho^{2m}(\\square_k,\\g_\\Pi^k)$\nand the operator $\\cL_\\e(\\om)$ is a sum of such single cell actions.\nSince the operators $\\cL_i$ are $\\Op_0$-bounded and symmetric, by  the Kato-Rellich theorem  the operator $\\Op_\\e(\\om)$ is self-adjoint for sufficiently small $\\e$.\n\n\n\n\n\nOur main aim is to study the behavior of the spectrum $\\spec(\\Op_\\e(\\om))$ of the operator $\\Op_\\e(\\om)$.\nThe later results will consider the asymptotic behaviour  for (very) small $\\e>0$.\nFor our first main result we merely require that $\\e$ is sufficiently small so that for all $\\omega$ the\nperturbed operator $\\Op_\\e(\\om)$ is self-adjoint.\n\n\\begin{theorem}\\label{th2.1}\nFor all sufficiently small $\\e$ there exists a closed set $\\Sigma_\\e$ such that\n\\begin{equation}\\label{2.9}\n\\spec(\\Op_\\e(\\om))=\\Sigma_\\e\\quad\\PP-a.s.\n\\end{equation}\nThe set $\\Sigma_\\e$ is equal to the closure of the union of spectra of periodic realizations of $\\Op_\\e(\\om)$, explicitly,\n\\begin{equation}\\label{2.10}\n\\Sigma_\\e=\\overline{\\bigcup\\limits_{N\\in \\NN} \\quad \\bigcup\\limits_{\\xi \\ \\text{is $2^N \\G$-periodic}} \\spec\\big(\\Op_\\e(\\xi)\\big)},\n\\end{equation}\nwhere the second union is taken over all sequences $\\xi: \\G\\to\\supp \\mu $, which are periodic  with respect to   the sublattice $2^N \\G:=\\{2^N q:\\, q\\in\\G\\}$.\n\\end{theorem}\nHere we adopt the following convention: In statements which are deterministic, i.e.~valid for all configurations\n$\\xi\\colon \\G \\to \\supp\\mu$ we will denote the sequences by $\\xi$, in statements which are probabilistic, e.g.~hold only for almost all configurations, we will use the symbol $\\omega$ for the configuration $\\omega\\colon \\G \\to \\supp\\mu$.\n\nThe next part of our results is devoted to the position of $\\inf \\Sigma_\\e$.\nMore precisely, we shall describe how $\\inf \\spec(\\Op_0)$ is shifted by the perturbation $\\cL_\\e(\\om)$.\nFirst we describe the spectrum of $\\Op_0$. Since this operator is periodic, we can employ a Floquet-Bloch decomposition to find $\\spec(\\Op_0)$. We introduce the Brillouin zone\n\\begin{equation*}\n\\square^*:=\\{\\theta\\in\\la e_1,\\ldots,e_{d_1}\\ra:\\, \\theta=\\theta_1^* e_1^*+\\ldots +\\theta_{d_1}^* e_{d_1}^*,\\, \\theta_i^*\\in[0,1)\\},\n\\end{equation*}\nwhere $e_1^*,\\ldots,e_{d_1}^*$ are the vectors in the linear span $S:=\\la e_1,\\ldots,e_{d_1}\\ra$ defined by the conditions\n\\begin{equation*}\n(e_i,e_j^*)_{\\mathds{R}^d}=2\\pi\\d_{ij},\n\\end{equation*}\nand $\\d_{ij}$ is the Kronecker delta.\nOn $\\square$ we introduce the operator\n\\begin{equation}\\label{2.11}\n\\Op_0(\\theta):\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|} (\\p + i \\theta)^\\a A_{\\a\\b} (\\p + i \\theta)^\\b \n\\end{equation}\nsubject to boundary conditions (\\ref{2.5}) and to periodic boundary conditions on $\\g_l$.\nHere, $\\iu$ is the imaginary unit and for a given $\\a=(\\a_1,\\ldots,\\a_d)\\in \\mathds{Z}_+^d$, $\\theta=(\\theta_1,\\ldots,\\theta_d)\\in S$, the symbol $(\\p + i \\theta)^\\a$ stands for the differential expression\n\\begin{equation*}\n\\left(\\frac{\\p\\,}{\\p x_1}+ i\\theta_1\\right)^{\\a_1} \\left(\\frac{\\p\\,}{\\p x_2}+ i\\theta_2\\right)^{\\a_2}\\cdot\\dots\\cdot\\left(\\frac{\\p\\,}{\\p x_d}+ i\\theta_d\\right)^{\\a_d}.\n\\end{equation*}\nThe spectrum of $\\Op_0$ is given by the identity \\cite[Ch. 4, Sect. 4.5, Thm. 4.5.1]{K93}\n\\begin{equation}\\label{2.12}\n\\spec(\\Op_0)=\\bigcup\\limits_{\\theta\\in\\square^*} \\spec(\\Op_0(\\theta)).\n\\end{equation}\nWe assume that $\\inf\\spec(\\Op_0(\\theta))$ is a discrete eigenvalue for all $\\theta\\in\\square^*$.\nWe denote this eigenvalue by $E_0(\\theta)$.\nThe function $\\theta\\mapsto E_0(\\theta)$ is $\\square^*$-periodic and continuous in $\\overline{\\square^*}$, a compact set.\nConsequently it attains the global minimum at some point $\\theta_0\\in\\square^*$:\n\\begin{equation}\\label{2.14}\n\\L_0:=E_0(\\theta_0)=\\min\\limits_{\\square^*} E_0(\\theta).\n\\end{equation}\nThere is a positive constant  $c_1$ independent of $\\theta\\in\\square^*$ such that\n\\begin{equation}\\label{2.13}\n\\dist\\Big(E_0(\\theta),\\spec(\\Op_0(\\theta))\\setminus\\{E_0(\\theta)\\}\\Big)\\geqslant c_1,\n\\end{equation}\nAgain by compactness and formula (\\ref{2.12})\n\\begin{equation}\\label{2.15}\n\\inf\\spec(\\Op_0)=\\L_0.\n\\end{equation}\nThe eigenfunctions of $\\Op_0(\\theta_0)$ associated with $E_0(\\theta_0)$ and orthonormalized in $L_2(\\square)$ are denoted by $\\psi_0^{(j)}$, $j=1,\\ldots,n$.\nWe choose these eigenfunctions so that the matrix with the entries\n$(\\cL_1 e^{\\iu \\theta_0 x}\\psi_0^{(i)},e^{\\iu \\theta_0 x}\\psi_0^{(j)})_{L_2(\\square)}$ is diagonal.\nThis  is possible by the theorem on simultaneous diagonalization of two quadratic forms in a finite-dimensional space.\n\n\nConsider the $2n $ numbers\n\\begin{equation*}\ns_-(\\cL_1 \\psi_0^{(i)},\\psi_0^{(i)})_{L_2(\\square)},\\quad i=1,\\ldots,n, \\quad s_+(\\cL_1 \\psi_0^{(i)},\\psi_0^{(i)})_{L_2(\\square)},\\quad i=1,\\ldots,n.\n\\end{equation*}\nAssume that the minimal value is attained at an index $i=i_0$ and for $s_*\\in\\{s_-,s_+\\}$. We denote\n\\begin{equation}\\label{2.16}\n\\L_1:=(\\cL_1e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)},\\quad \\psi_0:=\\psi_0^{(i_0)}.\n\\end{equation}\nBy $\\psi_1$ we denote the unique solution to the equation\n\\begin{equation}\\label{2.17}\n(\\Op_0(\\theta_0)-\\L_0) \\psi_1=-e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}\\psi_0+\\L_1\\psi_0\n\\quad\n\\perp \\ker  (\\Op_0(\\theta_0)-\\Lambda_0)\n\\end{equation}\nsuch that this solution is orthogonal to $\\psi_0^{(j)}$, $j=1,\\ldots,n$. We let\n\\begin{equation}\\label{2.18}\n\\L_2:=(\\cL_1e^{\\iu \\theta_0 x}\\psi_1,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} +(\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}.\n\\end{equation}\nOur next main result reads as follows.\n\\begin{theorem}\\label{th2.2}\nFor any minimizing triple $\\theta_0\\in\\square^*$, $i_0\\in\\{1, \\ldots,n\\}$, $s_*\\in\\{s_-,s_+\\}$\nas defined above, there exist $C,\\e _0 \\in(0,\\infty)$ such that for all $\\e \\in [0, \\e _0]$\nand for all sequences $\\xi\\colon \\Gamma \\to [s_-,s_+]$\n\\begin{equation}\\label{2.19}\n\\spec(\\Op_\\e(\\xi))\\subseteq\\big\\{\\l\\in\\mathds{R}:\\, \\dist(\\l,\\spec(\\Op_0))\\leqslant C\\e(|\\l|+1)\\big\\}.\n\\end{equation}\nFor the bottom of the almost sure spectrum $\\Sigma_\\e$ the estimate\n\\begin{equation}\\label{2.20}\n\\Sigma_\\e\n\\leqslant \\L_0+\n\\e s_* \\L_1 + \\e^2 s_*^2 \\L_2 + \\frac{\\e^3 s_*^3 \\L_3(\\e s_*)}{1+\\e^2 s_*^2 \\|\\psi_1\\|_{L_2(\\square)}^2}\n\\end{equation}\nholds true, where $\\L_3 \\colon \\RR \\to \\RR$ is a continuous function, in particular, uniformly  bounded on compact intervals.\n\\end{theorem}\nIn fact, one can show the following explicit representation for $\\L_3(t)$:\n\\begin{equation}\\label{2.21}\n\\begin{aligned}\n\\L_3(t)=  &-(\\L_1+t\\L_2)\\|\\psi_1\\|_{L_2(\\square)}^2\n+2 \\RE ( \\cL_2 e^{\\iu \\theta_0 x}\\psi_0,  e^{\\iu \\theta_0 x}\\psi_1)_{L_2(\\square)}\\\\\n & + \\big( (\\cL_1+t\\cL_2) e^{\\iu \\theta_0 x}\\psi_1,  e^{\\iu \\theta_0 x}\\psi_1\\big)_{L_2(\\square)}\n\\\\\n&+\\big(\\cL_3(t) e^{\\iu \\theta_0 x}\\psi_0 (\\psi_0+t\\psi_1),e^{\\iu \\theta_0 x}(\\psi_0+t\\psi_1)\\big)_{L_2(\\square)}.\n\\end{aligned}\n\\end{equation}\n\n\n\\begin{remark}\\label{rm2.2}\nIf we have several minimizing triples $\\{\\tht_0,i_0,s_*\\}$, estimate (\\ref{2.20}) is valid for each of them.\nIn this situation one should choose a triple, for which $s_* \\L_1$ is minimal.\nIf all such quantities are same, then one should minimize $s_*^2\\L_2$.\n\\end{remark}\n\n\n\nFor many types of periodic operators $\\Op_0$ and generic perturbations $\\cL_1$ the coefficient $\\L_1$ does not vanish,\nand one sees a linear shift of $\\Sigma_\\e$. Let us consider the case $\\L_1=0$.\nWe will show that in this case, for many models,\nthe almost sure spectrum $\\Sigma_\\e$ must expand at least quadratically\n ---  i.e.~cubic, quartic, or weaker expansions cannot occur.\n\n\\begin{corollary}\\label{c:1}\nAssume that $\\L_1=0$ and $\\cL_2$ is a non-positive operator, cf.~(\\ref{2.6}) and (\\ref{2.16}).\nThen\n\\[\n\\L_2 \\leqslant - c_1 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\]\nwhere $c_1$ is the strictly positive spectral gap in (\\ref{2.13}).\n\\end{corollary}\n\n\n\\begin{remark} \\label{r:1}\nThus the question arises, in what situations we can ensure that $\\psi_1$ does not vanish,\ni.e.~that $e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}\\psi_0$ is not the zero vector.\nThis is for instance the case if $\\Op_0$ is the pure Laplacian and $\\cL_1$ a multiplication operator $V$.\nIn this case $\\theta_0=0$, $\\Op_0(0)$ is the Laplacian with periodic boundary conditions, $\\psi_0$\nis the normalized constant function, so that $V\\psi$ is only identically zero if $V$ itself is trivial.\nMore generally, $\\Op_0$ can be any operator satisfying a unique continuation property for eigenfunctions\nor the Harnack inequality for the ground state.\n\nLikewise, $\\psi_1$ cannot vanish if $\\Op_0$ satisfies the Harnack inequality and\n$e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}$ is a positivity preserving operator.\\footnote{An $L_2$-function is called positive if $f$ is nonnegative almost everywhere and does not\nvanish identically. A self-adjoint operator $A$ on an $L_2$-space is called positivity preserving if $Af$ is positive\nwhenever $f$ is in the domain of $A$ and positive.}\n\\end{remark}\n\n\n\nOur next result provides a lower bound for $\\inf\\Sigma_\\e$.\nFirst we need to introduce additional notations and assumptions:\nGiven $u,v\\in \\Ho^{2m}(\\square,\\g_\\Pi)$, by integration by parts it is easy to convince oneself that\n\\begin{equation}\\label{2.23}\n\\begin{aligned}\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} & (-1)^{|\\a|} \\big( (\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u,v\\big)_{L_2(\\square)}\n\\\\\n&=\\sum\\limits_{j=1}^{m} (\\cB_{2m-j}u,\\cB_{j-1} v)_{L_2(\\g_l)}\n+ \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u, (\\p + \\iu \\theta_0)^\\a v\\big)_{L_2(\\square)},\n\\end{aligned}\n\\end{equation}\nwhere $\\cB_j=\\cB_j(x,\\p)$ are linear differential operators of order at most $j$ with continuous coefficients defined on $\\overline{\\g_l}$.\nSince the functions $A_{\\a\\b}$ are $\\square$-periodic, the above formula remains true if we replace $\\square$ by $\\square_k$.\nWe make the following assumption:\n\n\\begin{enumerate}\\def\\theenumi{(A\\arabic{enumi})}\n\\item \\label{A1} There are real-valued functions  $b_j\\in C(\\overline{\\p\\square})$ such that\n\\begin{equation}\\label{2.25}\nb_j=\\frac{\\cB_{2m-j}\\psi_0}{\\cB_{j-1}\\psi_0} \\chi_{\\{\\cB_{j-1}\\psi_0\\neq0\\}}\n\\quad \\text{ on} \\quad  \\g_l,\\quad j=1,\\ldots,m,\n\\end{equation}\nwhere $\\chi_\\natural$ is the characteristic function of a set $\\natural$.\nAssume also that the coefficients of the boundary operators\n$\\cB_{2m-j}-b_j\\cB_{j-1}$ belong to $C^j(\\g_l)$.\n\\end{enumerate}\n\nOn $\\Ho^m(\\square,\\g_l)$  we introduce the sesquilinear form\n\\begin{equation}\\label{2.26}\n\\widehat{\\fm}_0(u,v):= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u, (\\p + \\iu \\theta_0)^\\a v\\big)_{L_2(\\square)} + \\sum\\limits_{j=1}^{m} (b_j \\cB_{j-1}u,\\cB_{j-1}v)_{L_2(\\g_l)}.\n\\end{equation}\nThanks to conditions (\\ref{2.1}) and assumption \\ref{A1}, this form is symmetric, lower-semibounded and closed.\nBy $\\widehat{\\Op}_0$ we denote the self-adjoint operator in $L_2(\\square)$ associated with the form $\\widehat{\\fm}_0$.\nThis is the operator\n\\begin{equation}\\label{2.27}\n\\widehat{\\Op}_0= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}(\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b,\n\\end{equation}\nin $L_2(\\square)$ subject to the Dirichlet condition on $\\g_\\Pi$ and to the condition\n\\begin{equation}\\label{2.28}\n(\\cB_{2m-j}-b_j \\cB_{j-1})u=0\\quad \\text{on}\\quad \\g_l.\n\\end{equation}\nThe domain of $\\widehat{\\Op}_0$ consists of the functions in $\\Ho^{2m}(\\square,\\g_{\\Pi})$\nsatisfying the above boundary conditions on $\\g_l$.\n\n\nIt is straightforward to check that $\\psi_0$ is an eigenfunction of $\\widehat{\\Op}_0$ associated with the eigenvalue $\\L_0$. Moreover, we make one more assumption\n\\begin{enumerate}\\def\\theenumi{(A\\arabic{enumi})}\\setcounter{enumi}{1}\n\\item \\label{A2} The bottom of the spectrum of $\\widehat{\\Op}_0$ is a simple eigenvalue  and equal to $\\L_0$.\n\\end{enumerate}\n\nLet $\\widehat{\\psi}_1\\in\\Dom(\\widehat{\\Op}_0)$ be the solution to the equation\n\\begin{equation*}\n\\big(\\widehat{\\Op}_0-\\L_0\\big)\\widehat{\\psi}_1=-e^{-\\iu \\theta_0 x}\\cL_1e^{\\iu \\theta_0 x}\\psi_0 + \\L_1\\psi_0\n\\end{equation*}\nsuch that this solution is orthogonal to $\\psi_0$ in $L_2(\\square)$. We denote\n\\begin{equation}\\label{2.29}\n\\widehat{\\L}_2:=(\\cL_2e^{ \\iu \\theta_0 x}\\psi_0,e^{ \\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} + (\\cL_1 e^{ \\iu \\theta_0 x}\\widehat{\\psi}_1,e^{ \\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}.\n\\end{equation}\n\nThe main result concerning a lower bound is provided by the following theorem.\n\n\\begin{theorem}\\label{th2.3}\nAssume \\ref{A1} and \\ref{A2}. For all sufficiently small $\\e\\geqslant 0$ the estimate\n\\begin{equation}\\label{2.30}\n\\inf\\Sigma_\\e\\geqslant\n\\L_0+\\e s_* \\L_1+\\e^2 s_*^2\\widehat{\\L}_\n - C\\e^3\n\\end{equation}\nholds true, where $C$ is a non-negative constant independent of $\\e$.\n\\end{theorem}\n\nAn immediate corollary of Theorems~\\ref{th2.2} and \\ref{th2.3} is that under Assumptions~\\ref{A1},~\\ref{A2} we have the asymptotics\n\\begin{equation}\\label{2.31}\n\\inf\\spec(\\Op_\\e(\\om))=\\L_0+\\e s_*\\L_1+O(\\e^2)\\quad\\PP-a.s.\n\\end{equation}\n{Moreover, if $\\widehat{\\L}_2=\\L_2$ for at least one of the minimizing triples $(\\theta_0,i_0,s_*)$, the asymptotics is even more precise}\n\\begin{equation}\\label{2.32}\n\\inf\\spec(\\Op_\\e(\\om))=\\L_0+\\e s_*\\L_1+\\e^2 s_*^2\\L_2+O(\\e^3)\\quad\\PP-a.s.\n\\end{equation}\nNote that the additional assumptions~\\ref{A1},~\\ref{A2} required for the lower bound (\\ref{2.30}) are not very restrictive.\nA simple example is provided by the  operator $\\Op_0$ in $\\Pi=\\mathds{R}^d$ in the case $A_{\\a\\b}=0$ for $|\\a|<m$ or $|\\b|<m$. Then the ellipticity condition in (\\ref{2.1}) implies that $(\\Op_0 u,u)_{L_2(\\mathds{R}^d)}\\geqslant 0$.\nThe lowest eigenvalue of the operator $\\Op_0(\\theta)$ attains its minimum $\\L_0=0$ for $\\theta=0$; the associated eigenfunction $\\psi_0$ is just a constant: $\\psi_0=|\\square|^{-\\frac{1}{2}} \\mathbf{1}$, where $\\mathbf{1}$ stands for the function being $1$ everywhere in $\\square$.  This eigenvalue is simple.\nAll the associated functions $b_j$ vanish identically  and the form $\\widehat{\\fm}_0$ is given by the formula\n\\begin{equation*}\n\\widehat{\\fm}_0(u,v)= \\sum\\limits_{ \\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|,|\\b|=m}} (A_{\\a\\b} \\p^\\b u, \\p^\\a v )_{L_2(\\square)}.\n\\end{equation*}\nThe quadratic form $\\widehat{\\fm}_0(u,u)$ is non-negative and its value vanishes only for the  function $\\psi_0$.\nIt means that in this case both Assumptions~\\ref{A1},~\\ref{A2} are satisfied.\nThe constant  $\\L_1$ is given by the formula\n\\begin{equation*}\n\\L_1=\\frac{1}{|\\square|} (\\cL_1\\mathbf{1},\\mathbf{1})_{L_2(\\square)}.\n\\end{equation*}\nThe number $s_*$ is fixed by the inequalities\n\\begin{equation*}\ns_*\\L_1\\leqslant s_+\\L_1,\\quad s_*\\L_1\\leqslant s_-\\L_1.\n\\end{equation*}\nThe asymptotics (\\ref{2.31}) holds true.\n\n\nIt should be also mentioned that we choose Dirichlet boundary condition (\\ref{2.5}) just for simplicity.\nThe type of  boundary condition is not used in our proofs and the Dirichlet condition can be replaced by\nany other boundary condition ensuring the self-adjointness of the operator $\\Op_0$ on an appropriate subspace of $W_2^{2m}(\\Pi)$.\nThe operators $\\cL_i$ are to be symmetric on functions in $W_2^{2m}(\\square)$ satisfying the boundary condition on $\\g_\\Pi$ involved in the definition of $\\Op_0$. Then the main results remain true.\n\n\n\n\\section{Examples}\n\nIn this section we discuss examples of unperturbed operators $\\Op_0$ and perturbations $\\cL_\\e$.\nFirst of all we stress that\nall the examples we discuss below are considered in an arbitrary periodic domain $\\Pi$.\nClassical examples of the differential operator $\\Op_0$ are Schr\\\"odinger operators\n\\begin{equation*}\n\\Op_0=-\\D+V(x),\n\\end{equation*}\nmagnetic Schr\\\"odinger operators\n\\begin{equation*}\n\\Op_0=(\\iu\\nabla+A)^2,\n\\end{equation*}\noperators with a variable metric\n\\begin{equation*}\n\\Op_0=-\\sum\\limits_{i,j=1}^{d} \\frac{\\p}{\\p x_i } A_{ij} \\frac{\\p}{\\p x_j},\n\\end{equation*}\nor the bi-Laplacian\n\\begin{equation*}\n\\Op_0=\\Delta^2.\n\\end{equation*}\nIn the above examples all the coefficients $V$, $A$, $A_{ij}$ are supposed to be periodic  with respect to   $\\G$ and smooth enough.\n\n\nOne more class of examples are Schr\\\"odinger operators with $\\d$-interaction on a periodic system of disjoint bounded manifolds. Namely, let $M_0\\subset \\square$ be a given $C^3$-manifold\n{of codimension one without boundary}.\nBy $M$ we denote the periodic system of manifolds obtained by the shifts of $M_0$ along $\\G$. We consider the operator\n\\begin{equation}\\label{5.1}\n\\Op_0=-\\D+V\\quad \\text{in}\\quad \\Pi\n\\end{equation}\nsubject to the Dirichlet boundary condition on $\\p \\Pi$ and to the boundary conditions\n\\begin{equation}\\label{5.2}\n[u]_M=0,\\quad \\left[\\frac{\\p u}{\\p\\nu}\\right]_M=b u|_{M}=0,\n\\end{equation}\nwhere $[\\cdot]_M$ denotes the jump of a function on $M$, $\\nu$ is the normal vector to $M$ and $b\\in C^2(M)$ is a given function periodic  with respect to   lattice $\\G$.\nOf course, the above introduced operator does not fit our assumptions\nsince its domain is not $\\Ho^2(\\Pi,{\\p\\Pi})$. But it was shown in \\cite[Sect. 8, Ex. 5]{Bo07-AHP}, \\cite[Sect. 8, Ex. 5]{Bo07-MPAG}\nthat there exists an explicit unitary transformation that reduces operator (\\ref{5.1}), (\\ref{5.2}) into a self-adjoint differential operator with domain $\\Ho^2(\\Pi,{\\p\\Pi})$.\nThis new operator has the same spectrum as the operator introduced in (\\ref{5.1}) and  (\\ref{5.2}).\n\n\nThe class of examples of operators $\\cL_i$  is wider than for the operator $\\Op_0$.\nThe reason is that the operators $\\cL_i$ are not necessarily differential ones.\nOf course, symmetric differential operators are standard examples for $\\cL_i$:\n\\begin{equation*}\n\\cL_i=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (-1)^{|\\a|}\\p^\\a B_{\\a\\b}\\p^b,\n\\end{equation*}\nwhere $B_{\\a\\b}$  are given functions  defined on $\\square$ satisfying the conditions:\n\\begin{equation*}\n B_{\\a\\b}\\in W_\\infty^{|\\a|}(\\square),\n\\quad \\overline{B_{\\b\\a}}=B_{\\a\\b}.\n\\end{equation*}\nThe values on $\\g_l$ are to be chosen so that by the integration by parts we get\n\\begin{equation*}\n \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (-1)^{|\\a|} (\\p^\\a B_{\\a\\b}\\p^b u,v)_{L_2(\\square)}=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} ( B_{\\a\\b}\\p^b u, \\p^\\a v)_{L_2(\\square)},\\quad u,v\\in\\Ho^{2{m}}(\\square,\\g_\\Pi),\n\\end{equation*}\nand all the boundary integrals over $\\g_l$ vanish. In particular, this\nincludes potential and magnetic fields. In this case our results are applicable to the perturbation by a random potential and a random magnetic field.\n\nOur next example is an integral operator:\n\\begin{equation*}\n(\\cL_i u)(x)=\\int\\limits_{\\square} K(x,y)u(y)\\di y,\n\\end{equation*}\nwhere the kernel $K(x,y)$ is such that $\\cL_i$ is a symmetric operator. A sufficient condition is, for instance,\n\\begin{equation*}\nK(x,y)=\\overline{K(y,x)}\\quad\\text{and}\\quad \\int\\limits_{\\square\\times\\square} |K(x,y)|^2 \\di x\\di y<\\infty.\n\\end{equation*}\nAlso, more complicated examples like integro-differential operators or pseudodifferential operators are possible.\n\n\nThe next class of examples concerns the situation where a considered operator fits our assumptions\nafter certain transformations.\nHere the first example is the above mentioned $\\d$-interaction, see (\\ref{5.1}), (\\ref{5.2}). As $\\cL_i$, we can also choose a $\\d$-interaction of such type.\nOne just needs to replace the second boundary condition in (\\ref{5.2}) by\n\\begin{equation*}\n\\left[\\frac{\\p u}{\\p\\nu}\\right]_{M_k} + \\e \\omega_k b u\\big|_{M_k}=0,\n\\end{equation*}\nwhere $M_k$ is the shift of $M_0$ by $k\\in\\G$. Then we deal with a random $\\d$-potential.\nAnd again by applying the approach of \\cite[Sect. 8, Ex. 5]{Bo07-AHP}, \\cite[Sect. 8, Ex. 5]{Bo07-MPAG}\nwe can reduce the operator with  a $\\d$-interaction to a differential operator with the same spectrum.\nThis differential operator satisfies the representation (\\ref{2.8a}).\n\n\nThe next example is a small random deformation of a boundary. There are various ways how to introduce it, we dwell only on one of them.\nLet $\\nu$ be the outward normal to $\\p\\Pi$, $\\z$ be the distance measured along $\\nu$ and $\\rho\\in C^2(\\g_\\Pi)$ be a given function vanishing in the vicinity of $\\overline{\\g_l}\\cap\\p\\Pi$.\nWe introduce a domain $\\Pi_\\e(\\om)$ whose boundary is defined as\n\\begin{equation*}\n\\p\\Pi_\\e(\\om):=\\{x:\\, \\theta=\\e G(\\om,x)\\},\\quad G(\\om,x):=\\sum\\limits_{k\\in\\G} \\omega_k \\rho(x-k).\n\\end{equation*}\nIn the domain $\\Pi_\\e(\\om)$ we consider the operator\n\\begin{equation*}\n\\Op_\\e(\\om):=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}\\p^\\a A_{\\a\\b} \\p^\\b\n\\end{equation*}\nsubject to the Dirichlet condition with coefficients $A_{\\a\\b}$ satisfying (\\ref{2.1}). As in \\cite[Sect. 8, Ex. 4]{Bo07-MPAG},\none can construct a mapping of the domain $\\Pi_\\e(\\om)$  onto $\\Pi$ and a unitary transformation of $\\Op_\\e(\\om)$\nsuch that\nthe transformed  perturbed operator satisfies representation (\\ref{2.8a}) and has the same spectrum.\n\nOur next two examples are borrowed from \\cite{BKSh15}.\nLet $\\Wl=\\Wl(x)$ be a continuous compactly supported function defined on $\\mathds{R}^d$, and $\\Ws=\\Ws(x,\\z)$, $\\z=(\\z_1,\\ldots,\\z_d)$,\nstands for a function in $\\square\\times\\mathds{R}^{d}$\nbeing  1-periodic  with respect to   each of the variables $\\z_i$, $i=1,\\ldots,d$, having a zero mean\n\\begin{equation*}\n\\int\\limits_{(0,1)^{d}} \\Ws(x,\\xi)\\di\\xi=0\\quad \\text{for each} \\quad x\\in\\square,\n\\end{equation*}\nand compactly supported  with respect to   $x$:\n\\begin{equation*}\n\\supp \\Ws(\\cdot,\\xi)\\subseteq Q\\subset \\square \\quad\\text{for each}\\quad \\xi\\in\\mathds{R}^d,\n\\end{equation*}\nwhere $Q$ is a some fixed set. We assume the following smoothness for the function $\\Ws$:\n\\begin{equation*}\n\\frac{\\p^{|\\a|+|\\b|}\\Ws}{\\p x^\\a \\p\\xi^\\b} \\in C(\\square\\times\\mathds{R}^d),\\quad \\a,\\b\\in\\ZZ_+^d,\\quad |\\a|\\leqslant 3, \\quad |\\b|\\leqslant 1.\n\\end{equation*}\n We let\n\\begin{align*}\n&\\Wloc(x',\\e):=\\e^{-a} \\Wl\\left(\\frac{x'}{\\e}\\right), \\quad\n \\Wosc(x,\\e):=\\e^{-a} \\Ws\\left(x,\\frac{x}{\\e}\\right),\n \\\\\n & \\Wloc(x',0):=0,\\qquad \\Wosc(x',0):=0,\n\\end{align*}\nwhere $0\\leqslant a<1$ is a given number.\nThe potential $\\Wloc$ has {large} values due to the presence of the factor $\\e^{-a}$\nbut is localized on a set of diameter $\\e$. The potential $\\Wosc$ has also {large} values,\nis localized on set $Q$ and oscillates fast due to the presence of $\\frac{x}{\\e}$ in its definition.\n\n\nThe perturbed operators are introduced as\n\\begin{equation*}\n\\Op_\\e(\\om):=-\\D+ \\sum\\limits_{k\\in\\G} \\Wloc(\\cdot-k,\\e\\omega_k),\\quad \\Op_\\e(\\om):=-\\D+\\sum\\limits_{k\\in\\G} \\Wosc(\\cdot-k,\\e \\omega_k).\n\\end{equation*}\nIt was shown in  \\cite{BKSh15} that there exist unitary transformations reducing each of the above perturbed operators to operators satisfying representation (\\ref{2.8a}).\n\n\nWe also stress that any combination of the above perturbed operators is also covered by out general results. For instance, one can consider a combination of a random magnetic field with a random deformation of a boundary, a random integral operator and a $\\d$-interaction, a fast oscillating potential and a potential localized on a small set, etc.\n\n\n\n\n\\section{Proof of Theorem~\\ref{th2.1}}\nTo prove statement (\\ref{2.9}), we follow the ideas of \\cite{KirschM-82a}.\nIn accordance with Theorem~1 in that paper, it is sufficient to check the following conditions:\n\\begin{enumerate}\n\\item \\textsl{Measurability.}\n  For each $z\\in\\mathds{C}\\setminus\\mathds{R}$ the mapping $\\Om\\ni \\{\\xi_k\\}_{k\\in\\G}\\mapsto (\\Op_\\e(\\xi)-z)^{-1}$ defined on our product probability space $(\\Om,\\mathfrak{F},\\PP)$, where $\\mathfrak{F}:=\\bigotimes_\\G \\mathfrak{B}$ is the product $\\si$-algebra, is weakly measurable. Namely, for each $u,v\\in L_2(\\Pi)$ the mapping $\\{\\xi_k\\}_{k\\in\\G}\\mapsto \\big((\\Op_\\e(\\xi)-z)^{-1}u,v\\big)_{L_2(\\Pi)}$ is $\\PP$-measurable.\n\\item \\textsl{Homogeneity.}\n  There exists an index set $I$, a family of measure preserving transformations $T_i\\colon \\Om\\to\\Om$, $i\\in I$,\n  and a family of unitary operators $U_i$, $i\\in I$, on $L_2(\\Pi)$ satisfying for all $i \\in I$\n    \\begin{equation} \\label{eq:covariance}\n    \\Op_\\e( T_i \\xi)=U_i \\Op_\\e(\\xi) U_i^*.\n    \\end{equation}\n\\item \\textsl{Ergodicity.}\n    Any $A \\in \\mathfrak{F}$ such that $T_i^{-1}A=A$ for each $i\\in I$ satisfies  $\\PP(A)=0$ or $\\PP(A)=1$.\n\\end{enumerate}\nWe check the measurability following the argument of Proposition~6 in \\cite{KirschM-82a}.\nFirst we observe that by the continuity of $\\cL_3(t)$ in $t$, the scalar function $t\\mapsto (\\cL_3(t)u,v)_{L_2(\\square)}$ is measurable for each $u\\in\\Ho^{2m}(\\square,\\g_\\Pi)$, $v\\in L_2(\\square)$.\nThen we represent the resolvent of $\\Op_\\e(\\xi)$ as\n\\begin{align*}\n\\big(\\Op_\\e(\\xi)-z\\big)^{-1}=&(\\Op_0+\\cL_\\e(\\xi)-z)^{-1}\n\\\\\n=&(\\Op_0-z)^{-\\frac{1}{2}} \\big(I+(\\Op_0-z)^{-\\frac{1}{2}}\\cL_\\e(\\xi)(\\Op_0-z)^{-\\frac{1}{2}}\\big)^{-1} (\\Op_0-z)^{-\\frac{1}{2}}.\n\\end{align*}\n\nBy the lemma following Proposition~6 in \\cite{KirschM-82a} the operator $\\big(I+(\\Op_0-z)^{-\\frac{1}{2}}\\cL_\\e(\\xi)(\\Op_0-z)^{-\\frac{1}{2}}\\big)^{-1}$ is measurable and it implies the desired measurability of $\\Op_\\e(\\xi)$.\n\nTo show homogeneity, we let\n$I:=\\G$, $T_i(\\{\\xi_k\\}_{k\\in\\G}):=\\{\\xi_{k+i}\\}_{k\\in\\G}$, $U_i:=\\cS(i)$.\nIt is obvious that the operators $\\cS(i)$ are unitary and $\\cS^*(i)=\\cS^{-1}(i)=\\cS(-i)$. Employing definition (\\ref{2.8a}) of operator $\\Op_\\e(\\om)$ and the obvious identity $\\cS(k)\\cS(i)=\\cS(k+i)$, by straightforward calculations we see that\nCondition (\\ref{eq:covariance}) is satisfied.\nSince the marginal distributions of the product measure $\\PP$ are all equal, the mappings $T_i$ are measure preserving, namely, $\\PP(T_i^{-1}(A))=\\PP(A)$ for each $i\\in I$, $A\\in\\mathcal{F}$.\nMoreover, the product structure of $\\PP$ implies that the family $T_i, i\\in I$ is mixing, and in particular ergodic.\nThus there exists indeed a set $\\Sigma_\\e\\subseteq\\RR$ which is almost surely the spectrum of $\\Op_\\e(\\om)$.\nIn particular,  $\\Sigma_\\e$ is closed.\n\nIn order to prove (\\ref{2.10}), we adapt a similar proof in \\cite{KirschM-82b} (see Theorems~3,~4 and Lemma~2 in this paper).\nGiven $N\\in\\NN$, we denote\n\\begin{equation*}\n\\G_N:=\\{x\\in\\mathds{R}^d:\\, x=z_1 e_1 +\\ldots + z_{d_1}e_{d_1},\\,z_i\\in \\mathds{Z},\\, |z_i|\\leqslant N\\},\n\\end{equation*}\nand $\\Pi_N$ is the interior of $\\overline{\\bigcup\\limits_{k\\in\\G_N}\\square_k}$. For $\\xi,\\om\\in\\Omega$, $q\\in\\G$, we let\n\\begin{equation}\\label{3.0}\nD_N(\\xi,\\om,q):=\\left(\\sum\\limits_{k\\in\\G_N} \\|\\cL(\\e\\xi_k)-\\cS(q)\\cL(\\e\\omega_k) \\cS(-q)\\|_{\\Ho^{2m}(\\square_k,\\g_\\Pi^k)\\to L_2(\\square_k)}\\right)^{\\frac{1}{2}},\n\\end{equation}\nwhere $\\|\\cdot\\|_{X\\to Y}$ stands for the norm of an operator acting from Hilbert space $X$ into Hilbert space $Y$. We introduce the set\n\\begin{equation}\\label{3.4}\nB_\\xi:=\\left\\{\\om\\in\\Omega:\\, \\forall\\; N\\in \\NN \\  \\forall\\; p \\in \\NN \\\n\\exists\\;  q(N,p,\\xi)\\in\\G:\\, D_N(\\xi,\\om,q)<\\frac{1}{p} \\right\\}.\n\\end{equation}\n\n\\begin{lemma}\\label{lm3.1}\nFor each $\\xi\\in\\Om$ we have $\\PP(B_\\xi)=1$.\n\\end{lemma}\n\n\\begin{proof}\nWe follow the proof of Lemma~2 in \\cite{KirschM-82b}. We let\n\\begin{equation} \\label{eq:BxiNP}\nB_{\\xi,N,p}:=\\left\\{\\om\\in\\Omega:\\,\n\\exists\\; q(\\om)\\in\\G:\\, D_N(\\xi,\\om,q)<\\frac{1}{p}\\right\\}.\n\\end{equation}\nThe definition of $D_N$ implies $D_{N+1}(\\xi,\\om,p)\\geqslant D_N(\\xi,\\om,p)$ and hence $B_{\\xi,N+1,p}\\subseteq B_{\\xi,N,p}$.\nFurthermore $B_{\\xi,p+1}\\subseteq B_{\\xi,p}$.\nSet $B_{\\xi,p}:=\\bigcap\\limits_{N=1}^{\\infty} B_{\\xi,N,p}$\nand $B_\\xi:=\\bigcap\\limits_{p=1}^{\\infty} B_{\\xi,p}$.\nMonotone continuity of probability measures implies then\n\\begin{align*}\n& \\PP(B_{\\xi,p})=\\lim\\limits_{N\\to\\infty} \\PP(B_{\\xi,N,p}),\n\\quad \\PP(B_\\xi)=\n\\lim\\limits_{p\\to\\infty} \\PP(B_{\\xi,p}).\n\\end{align*}\nThus, it is sufficient to prove that $\\PP(B_{\\xi,N,p})=1$ for each $N,p\\in \\NN$.\nSince the set $B_{\\xi,N,p}$ is invariant under the shifts $\\{T_i\\}_{i\\in\\G}$,\nit is by ergodicity sufficient to prove that $\\PP(B_{\\xi,N,p})>0$.\nObviously\n\\begin{equation*}\nB_{\\xi,N,p}^*:=\\left\\{\\om\\in\\Om:\\, D_N(\\xi,\\om,0)<\\frac{1}{p}\\right\\}\n\\subseteq B_{\\xi,N,p}\n\\end{equation*}\n(just chose $q(\\omega)=0$).\nSo, it suffices to show $\\PP(B_{\\xi,N,p}^*)>0$ to complete the proof.\n\n\nThe definition of the operator $\\cL(t)$ yields\n\\begin{equation*}\n\\cL(t_1)-\\cL(t_2)=(t_1-t_2) \\cL_1 + (t_1^2-t_2^2) \\cL_2 + (t_1^3-t_2^3) \\cL_3(t_1) + t_2^3 \\big(\\cL_3(t_1)-\\cL_3(t_2)\\big).\n\\end{equation*}\nThis identity, definition (\\ref{3.0}) of $D_N$ and the continuity of operator $\\cL_3(t)$  in $t$ imply immediately\nthat $D_N(\\xi,\\om,0)$ tends to zero as $\\xi_i-\\omega_i\\to0$, $i\\in\\G_N$.\nHence, there exists $\\d=\\d(\\xi,p,N)$ such that $D_N(\\xi,\\om,0)<\\frac{1}{p}$ once $|\\xi_i-\\omega_i|<\\d$, $i\\in\\G_N$.\nIn other words,\n\\begin{equation*}\n\\left\\{\\om\\in\\Om:\\, |\\xi_i-\\omega_i|<\\d,\\, i\\in\\G_N\\right\\}\n\\subseteq B_{\\xi,N,p}^*\n\\end{equation*}\nSince  $\\G_N$ is finite, we see that the set on the right\nhas a positive measure, whenever all $\\xi_i, i \\in \\G_N$ are elements of $\\supp \\mu$.\nIt implies that $B_{\\xi,N,p}^*$ is of positive $\\PP$-measure.\n\\end{proof}\n\n\\begin{lemma}\\label{lm3.2}\nThe identity\n\\begin{equation*}\n\\Sigma_\\e=\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))\n=\\overline{\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))}\n\\end{equation*}\nholds true.\n\\end{lemma}\n\n\\begin{proof}\nWe adapt the proof of Theorem~3 in \\cite{KirschM-82b}. First of all, by (\\ref{2.9}) we have\n\\begin{equation*}\n\\Sigma_\\e\n\\subseteq \\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))\n\\end{equation*}\nLet us prove the opposite inclusion.\n\nWe fix $\\xi\\in\\Om$ and take $\\l\\in\\spec(\\Op_\\e(\\xi))$. By the Weyl criterion there exists a characteristic sequence $\\{\\vp_p\\}_{p\\in\\NN}\\subset \\Dom(\\Op_0)$ such that\n\\begin{equation}\\label{3.1}\n\\|\\vp_p\\|_{L_2(\\Pi)}=1,\\quad \\|(\\Op_\\e(\\xi)-\\l)\\vp_p\\|_{L_2(\\Pi)} \\leqslant \\frac 1 p, \\quad p\\in\\NN.\n\\end{equation}\nBy \\cite[Ch. III, Sect. 6, Lm. 6.3]{Ber} we have the estimate\n\\begin{equation*}\n\\|\\vp_p\\|_{W_2^{2m}(\\Pi)}\n\\leqslant C\\big (\\|(\\Op_0+\\iu)\\vp_p\\|_{L_2(\\Pi)}\n+|\\l|+1\\big  ),\n\\end{equation*}\nwhere $C$ is a constant independent of $p$ and $\\e$. Hence, the norm $\\|\\vp_p\\|_{W_2^{2m}(\\Pi)}$ is bounded uniformly in $p$, and the suprema\n\\begin{equation*}\n\\vp_p^*:=\\sup\\limits_{k\\in\\G} \\|\\vp_p\\|_{W_2^{2m}(\\square_k)},\n\\quad\n\\vp^*:=\\sup\\limits_{p\\in\\NN} \\vp_p^*\n\\end{equation*}\n{are} well-defined  and finite.\nPick $\\om$ from the full measure set\n$$\n\\Om^*:=B_\\xi\\cap \\{\\om\\in\\Om:\\, \\spec(\\Op_\\e(\\om))=\\Sigma_\\e\\}\n\\subseteq B_{\\xi,N_p,p}\n$$\nand\nlet us show that $\\psi_{p}:=\\cS(-q(N_p,p,\\om))\\vp_p$ is a characteristic sequence for $\\Op_\\e(\\om)$ at $\\l$.\nHere $N_p$ is a number fixed by the inequality\n\\begin{equation}\\label{3.2}\n\\|\\varphi_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\\leqslant \\frac{1}{p},\n\\end{equation}\nwhile  $q$ comes from definition (\\ref{eq:BxiNP}) of the set $B_{\\xi,N_p,p}$.\nEmploying definition (\\ref{3.4}) of the set $B_p$, (\\ref{3.1}), (\\ref{3.2}), by straightforward calculation we obtain\n\\begin{align*}\n\\|&(\\Op_\\e(\\om)-\\l)\\psi_{p}\\|_{L_2(\\Pi)} = \\|(\\Op_0 + \\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\l)\\vp_p\\|_{L_2(\\Pi)}\n\\\\\n&\\leqslant \\|(\\Op_\\e(\\xi)-\\l)\\vp_p\\|_{L_2(\\Pi)}  + \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi)}\n\\\\\n&\\leqslant  \\frac{1}{p} + \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi_{N_p})}\n\\\\\n&\\hphantom{\\leqslant}+ \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi\\setminus\\Pi_{N_p})}\n\\\\\n&\\leqslant \\frac{1}{p} + D_{N_p} (\\xi,\\om,q) \\vp^* + C\\|\\vp_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\n\\end{align*}\nwhere the constant $C$ is independent of $p$.\nSince the chosen $\\omega$ is in particular in $B_{\\xi,N_p,p}$ and by the choice (\\ref{3.2})\n\\begin{align*}\n\\frac{1}{p} + D_{N_p} (\\xi,\\om,q) \\vp^* + C\\|\\vp_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\n\\leqslant \\frac{1}{p} + \\frac{\\vp^* }{p} + \\frac{C}{p}.\n\\end{align*}\nHence, $\\psi_{p}$ is indeed a characteristic sequence for $\\Op_\\e(\\omega)-\\l$, i.e.~$\\l \\in\\spec(\\Op_\\e(\\omega))$\nfor any $\\omega \\in \\Omega^*$.\nTherefore, $\\spec(\\Op_\\e(\\xi))\\subseteq\\Sigma_\\e$ for each $\\xi\\in\\Om$. This completes the proof of the lemma.\n\\end{proof}\n\nIn view of the above lemma, we can obtain $\\Sigma_\\e$ as the union of the spectra $\\Op_\\e(\\xi)$ over all $\\xi\\in\\Om$.\nWe follow the proof of Theorem~4 in \\cite{KirschM-82b} to prove that this union can be taken over $2^k\\G$-periodic configurations $\\xi$ only.\n\nGiven $N\\in\\NN$, $\\xi\\in\\Om$, we introduce the $2^N\\G$-periodic sequence $\\xi^{(N)}\\in\\Om$ as follows: $\\xi_k^{(N)}=\\xi_k$, $k\\in\\G_{2^N}$, and for $k\\in\\G\\setminus\\G_{2^N}$ we define other terms of $\\xi^{(N)}$ by the periodic continuation. If we prove that $\\Op_\\e(\\xi^{(N)})$ converges to $\\Op_\\e(\\xi)$ as $N\\to\\infty$ in the strong resolvent sense, it will imply $\\spec(\\Op_\\e(\\xi))\\subseteq\\overline{\\bigcup\\limits_{N=1}^{\\infty} \\spec(\\Op_\\e(\\xi^{(N)}))}$, and hence,\n\\begin{equation*}\n\\overline{\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))}=\\overline{\\bigcup\\limits_{N\\in \\NN}\\bigcup\\limits_{\\xi \\ \\text{is $2^N \\G$-periodic}} \\spec\\big(\\Op_\\e(\\xi)\\big)},\n\\end{equation*}\nwhich will prove (\\ref{2.10}).\n\n\nThe desired strong resolvent convergence means that for each $f\\in L_2(\\Pi)$ the solution to the equation $\\big(\\Op_\\e(\\xi^{(N)})-\\iu\\big)u^{(N)}=f$ converges in $L_2(\\Pi)$ to the solution of the equation $\\big(\\Op_\\e(\\xi)-\\iu\\big)u=f$. These equations yield\n\\begin{equation*}\n\\big(\\Op_\\e(\\xi^{(N)})-\\iu\\big)(u-u^{(N)})=\\big(\\cL_\\e(\\xi^{(N)})-\\cL_\\e(\\xi)\\big)u.\n\\end{equation*}\nHence, by the definition of $\\xi^{(N)}$ and the estimate for the resolvent of a self-adjoint operator\n\\begin{equation*}\n\\|u-u^{(N)}\\|_{L_2(\\Pi)} \\leqslant \\big\\|\\big(\\cL_\\e(\\xi^{(N)})-\\cL_\\e(\\xi)\\big)u\\big\\|_{L_2(\\Pi)} \\leqslant C\\|u\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{2^N})},\n\\end{equation*}\nwhere $C$ is a constant independent of $u$ and $N$. Since $u\\in W_2^{2m}(\\Pi)$, we get $\\|u\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{2^N})}\\to0$ as $N\\to\\infty$\nand this completes the proof of Theorem~\\ref{th2.1}.\n\n\n\n\n\n\n\n\\section{Upper bound on $\\inf\\Sigma_\\e$: Proof of Theorem~\\ref{th2.2}.}\nIn order to prove (\\ref{2.19}), first  we estimate the resolvent of the unperturbed operator $\\Op_0$.\nSince this operator is self-adjoint, we have for any $\\l$ in the resolvent set\n\\begin{equation}\\label{4.1}\n\\|(\\Op_0-\\l)^{-1}\\|_{L_2(\\Pi)\\to L_2(\\Pi)}=\\frac{1}{\\dist(\\l,\\spec(\\Op_0))}.\n\\end{equation}\nWe rewrite the resolvent equation $(\\Op_0-\\l)u=f$ as\n\\begin{equation}\\label{4.2}\n(\\Op_0-\\iu)u=(\\l-\\iu)u+f.\n\\end{equation}\nSince the operator $(\\Op_0-\\iu)$ is invertible, by \\cite[Ch. I\\!I\\!I, Sect. 6, Lm. 6.3]{Ber} and (\\ref{4.1}), the solution of equation (\\ref{4.2}) satisfies the estimate\n\\begin{equation}\\label{4.3}\n\\|u\\|_{W_2^{2m}(\\Pi)} \\leqslant C\\|(\\l-\\iu)u+f\\|_{L_2(\\Pi)} \\leqslant \\frac{C(|\\l|+1)}{\\dist(\\l,\\spec(H_0))}\\|f\\|_{L_2(\\Pi)},\n\\end{equation}\nwhere $C$ is a positive constant independent of $\\l$ and $f$.\n\nConsider the resolvent equation for $\\Op_\\e(\\xi)$, $\\xi\\in\\Om$:\n\\begin{equation*}\n(\\Op_0+\\cL_\\e(\\xi)-\\l)u=f.\n\\end{equation*}\nWe can rewrite it as\n\\begin{equation*}\n\\big(I+\\cL_\\e(\\xi)(\\Op_0-\\l)^{-1}\\big)(\\Op_0-\\l)u=f,\n\\end{equation*}\nwhere $I$ is the identity mapping. Hence, once\n\\begin{equation}\\label{4.4}\n\\|\\cL_\\e(\\xi)(\\Op_0-\\l)^{-1}\\|<1,\n\\end{equation}\nthe resolvent of $\\Op_\\e$ is well-defined and  is given by the formula\n\\begin{equation*}\n(\\Op_\\e(\\xi)-\\l)^{-1}=(\\Op_0-\\l)^{-1} \\big(I+\\cL_\\e(\\xi)(\\Op_0\\l)^{-1}\\big)^{-1}.\n\\end{equation*}\nIn view of the definition of operators $\\cL_i$ and estimate (\\ref{4.3}), we see that inequality (\\ref{4.4}) is satisfied provided\n\n\\begin{equation}\\label{4.5} \\frac{C\\e(|\\l|+1)}{\\dist(\\l,\\spec(\\Op_0))}\\sup_{|t|\\leqslant t_0} \\|\\cL(t)\\|_{W_2^{2m}(\\square) \\to L_2(\\square)}<1,\n\\end{equation}\nwhere constant $C$ is the same as in (\\ref{4.3}). Hence, such $\\l$ are in the resolvent set of $\\Op_\\e(\\om)$.\nA contraposition of this statement yields (\\ref{2.19}).\n\n\nGiven $s\\in\\supp \\mu$, we denote by $\\xi^s$ the constant sequence $\\{s\\}_{k\\in\\G}$.\nIt follows from (\\ref{2.10}) that $\\Sigma_\\e\\supseteq\\spec(\\Op_\\e(\\xi^s))$, and hence, by the minimax principle\n\\begin{equation}\\label{4.6}\n\\inf\\Sigma_\\e\\leqslant \\inf\\spec(\\Op_\\e(\\xi^s))=\\inf\\limits_{\\genfrac{}{}{0 pt}{}{u\\in\\Dom(\\Op_\\e(\\xi^s))}{u\\not=0}} \\frac{\\big(\\Op_\\e(\\xi^s)u,u\\big)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\nLet $\\phi^\\e(x,s):=\\psi_0(x)+\\e s\\psi_1(x)$. Since both functions $\\psi_0$ and $\\psi_1$ satisfy periodic boundary conditions on $\\g_l$, we extend $\\phi_s^\\e$ periodically to $\\Pi$ keeping the same notation for the extension.\nBy $\\chi_p\\colon \\Pi \\to [0,1]$ we denote an infinitely differentiable function being one in $\\Pi_p$, vanishing in $\\Pi\\setminus \\Pi_{p+2}$ and satisfying the estimates\n\\begin{equation}\\label{4.7}\n\\left|\\frac{\\p^\\a \\chi_p}{\\p x^\\a}\\right| \\leqslant C_\\a\n\\quad \\text{in}\\quad \\overline{\\Pi_{p+2}\\setminus\\Pi_p},\n\\end{equation}\nwhere $C_\\a$ is a positive constant independent of $p$ and $x$.\nWe also suppose that $\\chi_p$ in fact depends only on the $d_1$ coordinates in $S$.\nMore precisely, for each pair $x,\\, \\widetilde{x}\\in\\Pi$ such that $x-\\widetilde{x}$ is orthogonal to $S$, we have $\\chi_p(x)=\\chi_p(\\widetilde{x})$.\n\nIn view of the above described properties of $\\phi^\\e$ and $\\chi_p$, the function\n\\begin{equation*}\nu_p^\\e(x):=e^{\\iu\\theta_0x}\\phi^\\e(x,s)\\chi_p(x)\n\\end{equation*}\nbelongs to the domain of $\\Op_0$ and therefore, $u_p^\\e\\in \\Dom(\\Op_\\e(\\xi^s))$. We choose $u_p^\\e$ as the test function in (\\ref{4.6}) to obtain\n\\begin{equation}\\label{4.8}\n\\inf\\Sigma_\\e\\leqslant \\frac{\\big(\\Op_\\e(\\xi^s)u_p^\\e,u_p^\\e\\big)_{L_2(\\Pi)}}{\\|u_p^\\e\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\n\n\nLet us calculate the right hand side of this inequality. It is clear that\n\\begin{equation}\\label{4.9}\n\\frac{\\big(\\Op_\\e(\\xi^s)u_p^\\e,u_p^\\e\\big)_{L_2(\\Pi)}}{\\|u_p^\\e\\|_{L_2(\\Pi)}^2}=\n\\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e \\chi_p,\\phi^\\e \\chi_p\\big)_{L_2(\\Pi)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\n\nIt follows from the definition of $\\phi^\\e$ that for each $k\\in\\G$ the identities\n\\begin{equation}\\label{4.10}\n\\begin{aligned}\n&\\|\\phi^\\e\\|_{L_2(\\square_k)}^2=1+\\e^2s^2\\|\\psi_1\\|_{L_2(\\square)}^2,\n\\\\\n&\\|\\phi^\\e\\|_{W_2^{2m}(\\square_k)}^2=\\|\\psi_0\\|_{W_2^{2m}(\\square)}^2\n+2\\e s\\RE(\\psi_0,\\psi_1)_{W_2^{2m}(\\square)} + \\e^2 s^2 \\|\\psi_1\\|_{W_2^{2m}(\\square)}^2\n\\end{aligned}\n\\end{equation}\n{hold true.}\nThese identities and the above described properties of $\\chi_p$ imply\n\\begin{align}\n&\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2=\\|\\phi^\\e\\|_{L_2(\\Pi_p)}^2 + \\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}^2,\\nonumber\n\\\\\n&\\|\\phi^\\e\\|_{L_2(\\Pi_p)}^2\\geqslant C p^{d_1},\n\\quad\n\\|\\phi^\\e\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}^2\\leqslant C p^{d_1-1}, \\label{4.11}\n\\end{align}\nwhere symbol $C$ stands for inessential constants independent of $p$, $\\e$ and $s$.\n\nThe boundedness of $\\cL_1, \\cL_1, \\cL_2, \\cL_3(t) : W_2^{2m}(\\square)\\to L_2(\\square)$ and (\\ref{4.7}) yield\n\\begin{equation}\\label{4.12}\n\\big\\|\\big(\\Op_0(\\theta_0) + e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x}\\big)\\phi^\\e \\chi_p \\big\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)} \\leqslant C \\|\\phi^\\e\\|_{W_2^{2m}(\\Pi_{p+2}\\setminus\\Pi_p)}\\leqslant C p^{d_1-1},\n\\end{equation}\nwhere $C$ is a constant independent of $p$, $\\e$ and $s$.\n\n\nIt is clear that\n\\begin{align}\\label{4.13}\n& \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e \\chi_p,\\phi^\\e \\chi_p\\big)_{L_2(\\Pi)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2}= T_p^1(\\e,s) + T_p^2(\\e, s),\n\\\\\n&T_p^1(\\e,s):= \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\Pi_p)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2})}^2},\\nonumber\n\\\\\n&T_p^2(\\e,s):= \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\tau_0 x})\\phi^\\e\\chi_p,\\phi^\\e \\chi_p \\big)_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2})}^2}.\n\\end{align}\nBy (\\ref{4.11}), (\\ref{4.12}) we obtain\n\\begin{equation}\\label{4.14}\n|T_p^2(\\e,s)|\\leqslant C p^{-1},\n\\end{equation}\nwhere the constant $C$ is independent of $p$, $\\e$, and $s$, and\n\\begin{equation}\\label{4.15}\n\\lim\\limits_{p\\to+\\infty}  T_p^1(\\e,s) = \\frac{\\big((\\Op_0(\\tau_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\square)}}{\\|\\phi^\\e\\|_{L_2(\\square)}^2}.\n\\end{equation}\nTogether with (\\ref{4.8}), (\\ref{4.9}) it yields\n\\begin{equation}\\label{4.16}\n\\inf \\Sigma_\\e\\leqslant \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\square)}}{\\|\\phi^\\e\\|_{L_2(\\square)}^2}.\n\\end{equation}\nEquation (\\ref{2.17}) and the eigenvalue equation for $\\psi_0$ imply that\n\\begin{align*}\n\\big(\\Op_0(\\theta_0) + e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x}\\big)\\phi^\\e=&\\L_0\\phi^\\e + \\e s \\L_1 \\psi_0\n\\\\\n&+ \\e^2 s^2 e^{-\\iu \\theta_0 x} (\\cL_2 + \\e s \\cL_3(\\e s) ) e^{\\iu \\theta_0 x} \\phi_\\e\n\\\\\n &+ \\e^2 s^2 e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x} \\psi_1.\n\\end{align*}\nSubstituting this identity into (\\ref{4.16}), we get\n\\begin{equation}\\label{4.17}\n\\begin{aligned}\n\\inf\\Sigma_\\e\\leqslant &\\L_0+ \\frac{1}{\\|\\phi^\\e\\|_{L_2(\\square)}^2} \\Big(\n\\e s\\L_1(\\psi_0,\\phi^\\e)_{L_2(\\square)} + \\e^2 s^2 (\\cL_2 e^{\\iu\\tht_0 x}\\phi^\\e, e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n\\\\\n&+\\e^2 s^2 (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n+\\e^3 s^3 (\\cL_3(\\e s) e^{\\iu\\tht_0 x}\\phi^\\e,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}.\n\\Big)\n\\end{aligned}\n\\end{equation}\nWe employ the identities\n\\begin{equation*}\n(\\psi_0,\\psi_1)_{L_2(\\square)}=0,\\quad \\|\\phi^\\e\\|_{L_2(\\square)}^2=1+\\e^2 s^2 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\end{equation*}\nand definitions (\\ref{2.16}), (\\ref{2.18}) of $\\L_1$, $\\L_2$\nto check that\n\\begin{equation*}\n \\e s\\L_1 (\\psi_0,\\phi^\\e)_{L_2(\\square)}=\\e s \\L_1\\|\\phi^\\e\\|_{L_2(\\square)}^2 - \\e^3 s^3 \\L_1 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\end{equation*}\nand\n\\begin{align*}\n(\\cL_2 e^{\\iu\\tht_0 x}\\phi^\\e,&e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)} + (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n\\\\\n=& \\L_2\\big(\\|\\phi^\\e\\|_{L_2(\\square)}^2 - \\e^2 s^2 \\|\\psi_1\\|_{L_2}^2\\big)\n+ 2\\e s \\RE (\\cL_2 e^{\\iu\\tht_0 x}\\psi_0,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)}\n\\\\\n&  + \\e^2 s^2 (\\cL_2 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)} + \\e s (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)}.\n\\end{align*}\nTogether with (\\ref{4.17}) it yields\n\\begin{equation*}\n\\inf \\Sigma_\\e \\leqslant \\L_0 + \\e s \\L_1 + \\e^2 s^2 \\L_2 + \\frac{\\e^3 s^3 \\L_3(\\e s)}{1+\\e^2 s^2 \\|\\psi_1\\|_{L_2(\\square)}^2}.\n\\end{equation*}\nAnd thanks to (\\ref{2.9}) it proves (\\ref{2.20}). The proof of Theorem~\\ref{th2.2} is complete.\n\n\\medskip\n\nIn this section we also prove Corollary~\\ref{c:1}.\n\n\\begin{proof}[Proof of Corollary \\ref{c:1}]\nSince {the function $e^{\\iu \\theta_0 x}\\psi_0(x)$}\nis in the domain of $\\cL_2$,\n\\begin{equation*}\n(\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} \\leqslant 0.\n\\end{equation*}\nFor  $\\L_1=0$ {we have}\n\\begin{align*}\n(\\cL_1e^{\\iu \\theta_0 x}\\psi_1,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}\n= &-(\\Op_0(\\theta_0) \\psi_1,\\psi_1)_{L_2(\\square)}\n= -(E_2-\\L_0) \\|\\psi_1\\|^2_{L_2(\\square)}\n\\\\\n\\leqslant &-c_1 \\|\\psi_1\\|^2_{L_2(\\square)},\n\\end{align*}\nwhere $E_2=\\spec(\\Op_0(\\theta))\\setminus\\{E_0(\\theta)\\}$ and $c_1$ the spectral gap.\nThus,\n\\[\n\\L_2\n\\leqslant - c_1 \\|\\psi_1\\|^2 + (\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}\n\\leqslant - c_1 \\|\\psi_1\\|^2.\n\\]\n\\end{proof}\n\n\n\\section{Lower bound on $\\inf\\Sigma_\\e$: Proof of Theorem~\\ref{th2.3}}\nLet $\\g_l^\\pm$ be a pair of opposite faces in $\\g_l$, namely,\n\\begin{equation*}\n\\g_l^-:=\\p\\square\\cap\\p\\square_{-e_i},\\quad \\g_l^+:=\\p\\square\\cap\\p\\square_{e_i}\n\\end{equation*}\nfor some $1\\leqslant i\\leqslant d_1$.\nWe recall that $e_1,\\ldots,e_{d_1}$ is the basis of lattice $\\G$, and $\\square_{\\pm e_i}$ is just $\\square_k$ with $k=\\pm e_i$. Let us show that\n\\begin{equation}\\label{4.18}\nb_j\\big|_{\\g_l^-}=-b_j\\big|_{\\g_l^+}\n\\end{equation}\nfor any choice of $1\\leqslant i\\leqslant d_1$.\nWe first observe that by the periodicity of the functions $A_{\\a\\b}$ and $\\psi_0$ we have\n\\begin{equation}\\label{4.18a}\n\\Big|\\cB_j\\psi_0\\big|_{\\g_l^-}\\Big|=\\Big|\\cB_j\\psi_0\\big|_{\\g_l^+}\\Big|.\n\\end{equation}\nFor each $u\\in \\Dom(\\Op_0)$\nwith compact support we have\n\\begin{align*}\n \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|} & \\int\\limits_{\\Pi} \\overline{\\psi_0}   (\\p + i \\theta)^\\a A_{\\a\\b} (\\p + i \\theta)^\\b u \\di x\n\\\\\n&= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}   \\int\\limits_{\\Pi} A_{\\a\\b} (\\p + i \\theta)^\\b u\\, \\overline{(\\p + i \\theta)^\\a\\psi_0}\\di x\n\\end{align*}\nand by (\\ref{2.23}) we also get\n\\begin{align*}\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} &(-1)^{|\\a|} \\int\\limits_{\\Pi} \\overline{\\psi_0}   (\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b u \\di x\n\\\\\n=& \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}\n \\\\\n &+ \\sum\\limits_{k\\in\\G}  \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}  \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b u, \\overline{(\\p + \\iu \\theta_0)^\\a\\psi_0}\\big)_{L_2(\\square_k)}\n\\\\\n=& \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}\n \\\\\n &+   \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}  \\int\\limits_{\\Pi} A_{\\a\\b} (\\p + i \\theta)^\\b u\\, \\overline{(\\p + i \\theta)^\\a\\psi_0}\\di x.\n\\end{align*}\nHence,\n\\begin{equation*}\n\\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}=0.\n\\end{equation*}\nSince $u$ is arbitrary and $\\psi_0$ is periodic, the above identity is possible only if for all $1\\leqslant i\\leqslant d_1$\n\\begin{align*}\n\\cB_{2m-j} \\cS(-k)u\\Big|_{\\g_l^-} \\overline{\\cB_j\\psi_0}\\Big|_{\\g_l^-}= & -\\cB_{2m-j} \\cS(-k-e_i)u\\Big|_{\\g_l^-} \\overline{\\cB_j\\cS(-e_i)\\psi_0}\\Big|_{\\g_l^-}\n\\\\\n=&-\\cB_{2m-j} \\cS(-k )u\\Big|_{\\g_l^+} \\overline{\\cB_j\\psi_0}\\Big|_{\\g_l^+}\n\\end{align*}\nfor each $k\\in\\G$. Dividing this identity by  $\\Big|\\cB_j\\psi_0\\Big|_{\\g_l^-}\\Big|^2=\\Big|\\cB_j\\psi_0\\Big|_{\\g_l^+}\\Big|^2$,\n cf.~(\\ref{4.18a}), and letting $u=\\psi_0$ in the vicinity of $\\square_k$, we arrive at (\\ref{4.18}).\n\n\nIt follows from (\\ref{4.18}) that for each\n{$u\\in\\Dom(\\Op_0)$}\n\\begin{equation*}\n\\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\cS(-k) u)_{L_2(\\g_l)} = 0.\n\\end{equation*}\nEmploying this identity and the minimax principle, we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\inf\\spec\\big(\\Op_\\e(\\xi)\\big)& = \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}} \\frac{\\fm_0(u,u) + (\\cL_\\e(\\xi)u,u)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}\n\\\\\n &=  \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}} \\frac{\\fm_0(e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u) + (\\cL_\\e(\\xi)e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}\n\\\\\n &=  \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\n\\frac{1}{\\|u\\|_{L_2(\\Pi)}^2} \\Big(\n \\fm_0(e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u) + ( \\cL_\\e(\\xi)e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u)_{L_2(\\Pi)}\n\\\\\n&\\hphantom{= \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\\frac{1}{\\|u\\|_{L_2(\\Pi)}^2} \\Big(}+ \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (b_j \\cB_{j-1}\\cS(-k)u,\\cB_{j-1}\\cS(-k)u)_{L_2(\\g_l)}  \\Big)\n\\\\\n&= \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\n\\frac{\\sum\\limits_{k\\in\\G} \\Big(\n\\widehat{\\fm}_0(u,u)  + \\big( \\cL(\\e\\xi_k)e^{\\iu\\theta_0x}\\cS(-k)u,e^{\\iu\\theta_0x}\\cS(-k)u\\big)_{L_2(\\square)}\n\\Big)}{\\sum\\limits_{k\\in\\G}\\|\\cS(-k)u\\|_{L_2(\\square)}^2}.\n\\end{aligned}\\label{4.20}\n\\end{equation}\nSince\n\\begin{equation*}\n\\Dom(\\Op_0)\n\\subseteq \\bigoplus\\limits_{k\\in\\G} \\Ho^{2m}(\\square_k,\\g_\\Pi^k):=\\left\\{u\\in L_2(\\Pi):\\, u\\big|_{\\square_k} \\in \\Ho^{2m}(\\square_k,\\g_\\Pi^k),\\,k\\in\\G\n\\right\\},\n\\end{equation*}\nby (\\ref{4.20}) we get\n\\begin{equation}\\label{4.21}\n\\begin{aligned}\n\\inf\\spec\\big(\\Op_\\e(\\xi)\\big)\n&\\geqslant  \\inf\\limits_{\\bigoplus\\limits_{k\\in\\G} \\Ho^{2m}(\\square_k,\\g_\\Pi^k)}\n\\frac{1}{\\sum\\limits_{k\\in\\G}\\|\\cS(-k)u\\|_{L_2(\\square)}^2}\n \\sum\\limits_{k\\in\\G}  \\Big(\n\\widehat{\\fm}_0(\\cS(-k)u, \\cS(-k)u)\n\\\\\n&\\hphantom{\\geqslant  \\inf\\limits_{\\bigoplus\\limits_{k\\in\\G}\\Ho^{2m}(\\square,\\g_\\Pi )} }+ \\big( \\cL(\\e\\xi_k)e^{\\iu\\theta_0x}\\cS(-k)u,e^{\\iu \\theta_0 x}\\cS(-k)u\\big)_{L_2(\\square)}\n\\Big)\n\\\\\n&\\geqslant \\inf\\limits_{[s_-,s_+]} \\l_\\e(s),\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n\\l_\\e(s):=\\inf\\limits_{\\Ho^{2m}(\\square,\\g_\\Pi) }    \\frac{\\widehat{\\fm}_0(u,u)  +\n\\big(\\cL(\\e s)e^{\\iu\\theta_0x} u, e^{\\iu\\theta_0x} u\\big)_{L_2(\\square)}\n}{\\|u\\|_{L_2(\\square)}^2}.\n\\end{equation*}\nBy the minimax principle, $\\l_\\e(s)$ is the bottom of the spectrum of the operator $$\\widehat{\\Op}_0 + e^{-\\iu\\theta_0 x}\\cL(\\e s)e^{\\iu\\theta_0 x}.$$ Assumption \\ref{A2} and the $\\widehat{\\Op}_0$-boundedness of $\\cL(\\e s)$ yield that $\\l_\\e(s)$ is a discrete eigenvalue of $\\widehat{\\Op}_0 + e^{-\\iu\\theta_0 x}\\cL(\\e s)e^{\\iu\\theta_0 x}$ and $\\l_\\e(s)\\to\\L_0$ as $\\e\\to+0$ uniformly in $s\\in[s_-,s_+]$. By regular perturbation theory one can easily construct the asymptotic expansion for $\\l_\\e(s)$:\n\\begin{equation}\\label{4.22}\n\\l_\\e(s)=\\L_0+\\e s\\widehat{\\L}_1 + \\e^2 s^2 \\widehat{\\L}_2 + O(\\e^3),\n\\end{equation}\nwhere the estimate for the error term is uniform in $s\\in[s_-,s_+]$, \\begin{equation*}\n\\widehat{\\L}_1=(e^{-\\iu\\theta_0 x}\\cL_1 e^{\\iu\\theta_0 x}\\psi_0,\\psi_0)_{L_2(\\square)}=\\L_1,\n\\end{equation*}\nand $\\widehat{\\L}_2$ is given by formula (\\ref{2.29}).\nThe asymptotics (\\ref{4.22}), estimate (\\ref{4.21}) and the definition of $\\l_\\e(s)$ imply\n(\\ref{2.30}).\nThe proof is complete.\n\n\n\\section*{Acknowledgments}\n\nThis work was initiated while the authors were at the Chair of Stochastics, Faculty of Mathematics, of the Technische Universit\\\"at Chemnitz. It was partially financially supported by the DFG through the project grant \\emph{Eindeutige-Fortsetzungsprinzipien und Gleichverteilungseigenschaften von Eigenfunktionen}. The research of D.B. was supported by the grant of Russian Science Foundation no. 14-11-00078.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:introduction}Introduction}\n\nFluid flow in porous media is studied for more than a century due to its high relevance to several engineering applications such as enhanced oil recovery~\\cite{green1998enhanced,farajzadeh2012foam,fraggedakis2015flow,sahimi2011flow}, filtration and separation~\\cite{herzig1970flow,tien1979advances,jaisi2008transport}, fermentation~\\cite{pandey2003solid,aufrecht2019pore}, soil sequestration~\\cite{schlesinger1999carbon}, energy storage~\\cite{duduta2011semi,sun2019hierarchical}, and food processing~\\cite{greenkorn1983flow}. In most cases, the fluids involved in these applications exhibit yield stress\/viscoplastic behavior~\\cite{bonn2009yield,balmforth2014yielding}. Therefore, understanding the conditions -- critical pressure drop and\/or stresses -- that lead to fluidization of yield-stress fluids in porous media can help boost the efficiency and lower the operational cost of several industrial applications.\n\nIn pressure-driven flows, the critical pressure drop $\\Delta P_c$ required to fluidize the yield-stress fluid and open the first channel~\\cite{chen2005flow,hewitt2016obstructed,waisbord2019anomalous} depends on the heterogeneous geometric characteristics and porosity $1-\\phi$ of the porous medium, where $\\phi$ is the volume fraction of the solid phase~\\cite{talon2013geometry,bauer2019experimental,chaparian2020complex}. Therefore, it is crucial to understand the relation between the yielding conditions and the structure of the porous medium, which will lead to predictive models for both the first open channel and $\\Delta P_c$.\n\nThe classical way to study yield-stress fluids is by solving the Cauchy momentum equations using viscoplastic constitutive relations, such as the Bingham and Herschel\u2013Bulkley models~\\cite{bird1987dynamics,huilgol2015fluid,saramito2016complex}. More recently, though, there is an increasing trend on using elastoviscoplastic~\\cite{saramito2007new} and kinematic hardening~\\cite{dimitriou2019canonical} models that originate from continuum mechanics~\\cite{gurtin2010mechanics,anand2020continuum}. The yield stress behavior, however, leads to an ill-defined problem that does not describe the stress distribution within the unyielded regions of the fluid~\\cite{balmforth2014yielding,saramito2017progress}. Common ways to resolve this problem is by using either optimization-~\\cite{hestenes1969multiplier,powell1978algorithms,glowinski2008lectures} or regularization-based methods~\\cite{papanastasiou1987flows,frigaard2005usage}. The former are accurate on predicting the yielded\/unyielded boundaries and the flow field~\\cite{dimakopoulos2013steady}, however, they are computationally expensive~\\cite{saramito2016damped} for conditions nearby the yield limit. Although the latter reduce the computational cost, they introduce non-physical parameters~\\cite{tsamopoulos2008steady,dimakopoulos2018pal} that lead to non-physical solutions, incorrect location of yield\/unyield boundaries, and inaccurate yield limits~\\cite{mitsoulis2017numerical,frigaard2005usage}. Here, we are interested in determining the statistics of the critical $\\Delta P_c$ for a yield-stress fluid in complex porous media. Thus, we need to use models that can predict accurately and efficiently $\\Delta P_c$ along with the first open channel.\n\nFluid flow in porous media is traditionally described through network models~\\cite{fatt1956network} that represent the complex geometric characteristics of the domain with spherical pore throats and cylindrical edges~\\cite{alim2017local,bryant1993network,blunt2001flow,blunt2013pore,stoop2019disorder}. In addition to their wide applicability in Newtonian fluids, network models have also been applied to describe $\\Delta P_c$ and the flow behavior with respect to the applied pressure drop in yield-stress fluids~\\cite{chen2005flow,frigaard2017bingham,liu2019darcy,talon2020effective}. When the relation between the local flow rate and the pressure drop is known, the network representation allows for the use of graph theoretic tools~\\cite{kharabaf1997invasion,chen2005flow,balhoff2012numerical,liu2019darcy} to quickly evaluate $\\Delta P_c$ and the flow response of the system. In general, though, the results of network viscoplastic models in complex porous media have been rarely compared and validated against those produced by solving the full fluid problem, and thus the conditions of their validity\/applicability are unknown. \n\nThe goal of the present work is to predict the first open channel for a yield-stress fluid in a complex porous medium along with the critical applied pressure drop required to open it. We develop a simple network model based on realistic porous media configurations, and use graph theoretical tools to study the statistics of yielding conditions in terms of the medium porosity. We validate our results against reported pressure-driven simulations of Bingham fluids in porous media. Finally, we discuss the relevance of our study to applications such as enhanced oil recovery and propose possible extensions.\n\n\\section{\\label{sec:modeling}Theory}\n\n\\begin{figure*}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.7\\textwidth]{fig1.pdf}\n    \\caption{(a) Typical configuration of a porous medium of length $L$ and porosity $1-\\phi$ that is made of non-overlapping monodisperse spheres of radius $R_s$, and its network representation. Across the domain, a macroscopic pressure gradient $\\Delta P\/L$ is applied to fluidize the yield-stress fluid. (b) Schematic of three network edges with different radii $r_i$ and lengths $l_i$ ($i=1,2,3$ ). The network representation includes the local geometric characteristics of the complex porous medium structure. (c) Characteristic pore-size distribution for the network shown in (a) as derived from the network model. (d) Typical shear stress response $\\tau$ as a function of the applied shear rate $\\dot{\\gamma}$ of a viscoplastic fluid with yield stress $\\tau_y$.}\n    \\label{fig:system}\n\\end{figure*}\n\n\\subsection{Network topology}\nWe are interested in the construction of realistic network models that capture the complex morphology of real porous media. The main scope of our work is to understand the statistics on the critical conditions that lead to fluidization in terms of the porosity $1-\\phi$ and topological characteristics of the medium. \n\nTo first approximation, we assume a porous medium that consists of monodisperse non-overlapping spheres\/disks of radius $R_s$ as shown in Fig.~\\ref{fig:system}(a). It is apparent that the structure of void space depends on the solid volume fraction defined as $\\phi=N_sV_s\/V_t$, where $N_s$ is the total number of spheres, $V_s$ is the volume of an individual sphere and $V_t$ the total volume of the system. We can use the given porous medium structure and create the network representation shown in the right panel of Fig.~\\ref{fig:system}(a).\n\nThe network consists of nodes and edges that span the entire medium, where its complex topological characteristics are encoded on the connectivity between them~\\cite{gostick2017versatile,khan2019dual}. Additionally, the local geometric characteristics of the porous medium are included on the length $l_i$ and radius $r_i$ of each individual edge, Fig.~\\ref{fig:system}(b). For demonstration, we show in Fig.~\\ref{fig:system}(c) the pore size distribution for the configuration of Fig.~\\ref{fig:system}(a). Details on the generation of porous media with monodisperse non-overlaping sphere, the construction of the network representation, and the choice of $r_i$ and $l_i$ are given in the Appendix of the paper.\n\n\\subsection{Yield-stress fluid in a network}\n\nYield-stress fluids are characterized by their solid-liquid transition when the Euclidean norm of the stress field exceeds the value of yield stress (Von Mises criterion)~\\cite{gurtin2010mechanics,hill1998mathematical}. The typical shear stress response in simple shear flow is shown in Fig.~\\ref{fig:system}(d), where for $\\dot{\\gamma}\\rightarrow0$ the shear stress reaches its critical value $\\tau\\rightarrow\\tau_y$. The most common constitutive relations used to describe yield-stress fluids are the Bingham and the Herschel-Bulkley models~\\cite{bird1987dynamics}. Both of them, though, have the same behavior for $\\dot{\\gamma}\\rightarrow0$. Therefore, it is sufficient to discuss only the Bingham model for a porous medium to understand the connection between the yielding conditions to the geometric and topological characteristics of the network.\n\nFor pressure-driven flows, the local flow rate $q_i$ of edge $i$ is described in terms of the local geometric properties $r_i,l_i$, the local pressure drop along the edge $\\Delta P_i$, and the yield stress $\\tau_y$ of the fluid as~\\cite{bird1987dynamics,liu2019darcy,frigaard2019background}\n\\begin{equation}\n\\label{eq:qflow}\nq_{i} = \\left\\{ \n\\begin{array}{cc}\n\\frac{r_i^4}{l_i} \\left(\\Delta P_{i}-\\frac{\\tau_y l_i}{r_i}\\right) &\\text{for} \\,\\, \\Delta P_{i}>\\frac{\\tau_y l_i}{r_i}\\\\\n 0   &\\text{ for} \\,\\, \\left\\lvert\\Delta P_{i}\\right\\rvert < \\frac{\\tau_y l_i}{r_i} \\\\\n\\frac{r_i^4}{l_i} \\left(\\Delta P_{i}+\\frac{\\tau_y l_i}{r_i}\\right) &\\text{ for} \\,\\, \\Delta P_{i}<-\\frac{\\tau_y l_i}{r_i}\\\\\n\\end{array} \n\\right.\n\\end{equation}\nNear the no-flow limit $q_i\\rightarrow0$, we see from Eq.~\\ref{eq:qflow} that $\\Delta P_i\\rightarrow \\tau_yl_i\/r_i$, and thus smaller in radius or longer in length channels require larger applied pressure drop to yield. Across the first open channel, we can calculate the total pressure drop across the medium to be $\\Delta P \\equiv \\sum_{i=1}^N \\Delta P_i = \\tau_y\\sum_{i=1}^N l_i\/r_i$, where $N$ is the total number of edges across the path. From this expression, we can see that the connectivity between the edges determines the first open channel in a real porous medium, and it corresponds to the path of `least resistance'. Thus, the problem of finding $\\Delta P_c$ can be formulated as finding the path of the minimum pressure drop as follows~\\cite{liu2019darcy}\n\\begin{equation}\n\\label{eq:least_res}\n    \\frac{\\Delta P_c}{\\tau_y} = \n    \\min_{C \\in \\mathcal{C}_{\\text{in-out}}}\n    \\sum_{i=1}^N \\frac{l_i}{r_i}\n\\end{equation}\nwhere $\\mathcal{C}_{\\text{in-out}}$ is the set of all paths between the corresponding boundaries. \n\nThe problem of Eq.~\\ref{eq:least_res} satisfies the principle of minimum dissipation rate and is valid near equilibrium~\\cite{kondepudi2014modern}. In particular, the entropy production for a pressure-driven flow is $\\sigma_D=q\\Delta P$~\\cite{de2013non}. Thus, for conditions near the solid-liquid transition where $q\\rightarrow0^+$, the minimum pressure drop path also minimizes $\\sigma_D$.\n\nTo solve Eq.~\\ref{eq:least_res}, we transform the generated network into a graph with edges that have weights equal to $l_i\/r_i$ and use the Dijkstra method~\\cite{dijkstra1959note} for directed graphs to determine the first open channel. This method is known to scale quadratically with the path length~\\cite{west1996introduction,bollobas2013modern}, and therefore for complex domains that lead to larger number of edges the computational cost increases. For a single porous medium configuration, however, the overall computational time to determine the first open channel is much lower (seconds to minutes) than that required to solve the full fluid flow problem using optimization methods (days to weeks)~\\cite{dimakopoulos2018pal,chaparian2020complex}. \n\n\\section{\\label{sec:results}Results}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.5\\textwidth]{fig2.pdf}\n    \\caption{Validation of the network model for the first open channel against simulations for a pressure-driven Bingham fluid in a complex porous medium with $R_s\/L=0.02$. (a) Simulation results of the open pathway for conditions near the critical pressure drop $\\Delta P_c$. The contour plot shows the magnitude of the local velocity, normalized with the maximum velocity across the channel. (b) Network model predictions for the first open channel. The cases of $\\phi=0.3$ and $\\phi=0.5$ are examined. It is clear that both the full Bingham fluid simulation and the network model predict the same location for the first open channel.}\n    \\label{fig:validation}\n\\end{figure}\n\n\\subsection{The first open channel}\nWhen the applied pressure drop approaches the critical value $\\Delta P \\rightarrow \\Delta P_c^+$, there exists a single open channel across the entire medium. Here, we test the validity of the proposed approach to determine the first open channel nearby when the solid-liquid transition occurs. For comparison, we solve the full flow field under pressure-driven conditions for a yield-stress fluid for the complex porous media shown in Fig.~\\ref{fig:validation}. We consider the cases of $\\phi=0.3$ and $\\phi=0.5$, respectively. All the lengths are normalized with the macroscopic length of the system $L$ and also $R_s\/L=0.02$. For simplicity, we use two-dimensional porous media, however, our approach is general and does not depend on the dimensionality of the problem.\n\nFigure~\\ref{fig:validation}(a) shows the normalized velocity magnitude that results from the solution of the Cauchy momentum equation for a Bingham fluid~\\cite{chaparian2020complex}. Also, Fig.~\\ref{fig:validation}(b) depicts the predictions for the first open channel after solving the minimization problem of Eq.~\\ref{eq:least_res}. In both cases, the network model is able to reproduce the results of the fluid problem for the first open path.\n\nNotably, in both cases of Fig.~\\ref{fig:validation}(a), where the full fluid flow problem is solved, we can see the existence of additional paths other than the one predicted by the network model. We justify this observation based on the fact that the fluid flow simulations are performed at extremely large, yet finite non-dimensional yield stress (i.e.~Bingham number; see \\cite{chaparian2020complex}). Indeed, we speculate that for simulations with $q\\rightarrow0^+$ (i.e.~infinite Bingham number), the secondary paths will eventually close and only the predicted one by the network model will be present. However, as it is clear from the Fig.~\\ref{fig:validation}(a), flow rate in those secondary paths are negligible and does not contribute to the leading order of the resistance.\n\n\\subsection{Critical pressure drop $\\Delta P_c$ and its statistics in complex porous media}\n\n\\begin{figure*}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.7\\textwidth]{fig3.pdf}\n    \\caption{Statistics on the predictions of the network model for 500 realizations per value of $\\phi$. (a) Critical pressure drop $\\Delta P_c$ as a function of the volume fraction $\\phi$ for different ratio of $R_s\/L$. $\\Delta P_c$ is normalized with the yield stress of the fluid $\\tau_y$. The error bars indicate the variance around the mean value. Inset -- Scaled critical pressure drop $\\Delta P_c R_s\/\\tau_y L$ in terms of $\\phi$. For non-overlapping spheres, the results for different $R_s\/L$ collapse in a master curve. (b) Probability density distributions for the normalized arc-length of the first open channel $L_c\/L-1$. The cases of $R_s\/L=0.02$ and $R_s\/L=0.08$ are shown, respectively, for $\\phi=0.1$ and $\\phi=0.5$. In general, increasing $\\phi$ leads to increase of $\\Delta P_c$ required to open the first channel, but also leads to a wide distribution of arc-lengths, which provide large uncertainty on the critical macroscopic pressure gradient $\\Delta P_c\/L_c$ required to yield the fluid in the porous medium.}\n    \\label{fig:result}\n\\end{figure*}\n\nIn addition to the first open path, the network model can predict the normalized critical pressure $\\Delta P_c$ required to yield the fluid in the porous medium. \n\n\\begin{table}[h]\n\\begin{ruledtabular}\n\\begin{tabular}{ c c c c}\n\\textrm{$\\phi$}&\n\\textrm{$\\frac{\\Delta P_c}{\\tau_y}$ Network}&\n\\textrm{$\\frac{\\Delta P_c}{\\tau_y}$ Simulations}&\n\\textrm{Rel. Error $\\%$}\\\\\n\\colrule\n$0.1$ & $22.16$ & $22.63$ & $1.87$\\\\\n$0.3$ & $61$ & $65.2$ & $6.43$\\\\\n$0.5$ & $142.16$ & $145.28$ & $2.05$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{\\label{tab:table_DPc}%\nPredicted normalized critical pressure drop $\\Delta P_c$ required to open the first open channel for the configurations shown in Fig.~\\ref{fig:validation}. We show both the predictions of the network model and the full simulation results, along with the relative error $\\frac{\\Delta P_{c}^{\\text{sim}}-\\Delta P_{c}^{\\text{net}}}{\\Delta P_{c}^{\\text{sim}}}$. The network model is adequate on predicting both the first open channel and the critical pressure drop required to open it.\n}\n\\end{table}\n\nIn table~\\ref{tab:table_DPc} we show the predictions of the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ for both the network model and the full simulations. For $\\phi=0.3$ and $0.5$ we consider the configurations shown in Fig.~\\ref{fig:validation}. In all cases, it is clear that the relative difference of $\\Delta P_c$ between the two models never exceeds $6.5\\%$. The negligible difference can be justified by the fact that discussed above about the secondary paths, or\/and the simplification of the geometric characteristics of the medium by its network representation. We conclude, though, that network-based models are adequate to predict both the first open path and the critical pressure drop required to open it. \n\nThe validation of the model allows us to predict $\\Delta P_c$ as a function of $\\phi$ and the geometric characteristics of the system. For a porous medium made of non-overlapping disks, we control the microstructure characteristics by changing the ratio $R_s\/L$. For each combination of $\\phi$ and $R_s\/L$, we generate 500 realizations for gathering the statistics of $\\Delta P_c$. \n\nFigure~\\ref{fig:result}(a) shows the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ in terms of $\\phi$. The colored lines indicate different value of $R_s\/L\\in[0.02,0.1]$, while the error bars correspond to the variance of the statistical sample. For all $R_s\/L$ the normalized pressure drop increases with increasing $\\phi$. This behavior is expected as the radius $r_i$ of each edge decreases monotonically as $\\phi$ increases~\\cite{torquato2002random}. Additionally, we observe that decreasing the ratio $R_s\/L$ leads to increase in $\\Delta P_c$, in qualitative agreement with experiments~\\cite{waisbord2019anomalous}. The reason for this behavior can be explained by examining the arc-length (defined as $L_c=\\sum_i^N l_i$) for the first open channel. \n\nFigure~\\ref{fig:result}(b) illustrates the histogram of $L_c$ for $\\phi=0.3$ and $0.5$ as well as $R_s\/L=0.02$ and $0.08$. In both cases we observe that increasing solid volume fraction leads to an increase in the total length of the first open path. While large $\\phi$ results in almost similar behavior for both $R_s\/L$, it is apparent that decreasing the size of the spheres results in more tortuous paths, even for low solid volume fraction. \n\nDimensional analysis on Eq.~\\ref{eq:least_res} indicates that $\\Delta P_c$ follows a simple scaling with $R_s\/L$. In particular, by rescaling the local edge radius $r_i$ with the sphere radius $R_s$, we find $\\Delta \\widetilde{P}_c=\\Delta P_c R_s\/\\tau_y L = \\sum_{i=1}^N \\left(l_i\/L\\right)\\left(R_s\/r_i\\right)$. The inset in Fig.~\\ref{fig:result}(a) shows the rescaled form of the critical pressure drop for all $R_s\/L$, were for non-overlapping disks, a master curve exists for all the examined values of $\\phi$. The dependence of $\\Delta \\widetilde{P}_c$ with $\\phi$, the effect of the pore microstructure, as well as the physical mechanism for the behavior of the tortuosity $\\tau=L_c\/L$ will be presented in a future work.\n\n\\section{\\label{sec:discussion}Discussion}\n\nWe find that network models provide an efficient and accurate way to model the fluidization conditions in porous media. They are also accurate in locating the first open channel, which is equivalent to the path of least resistance through the entire medium.\n\nOur results on the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ can be used to design porous media systems with the desired flow properties. In applications such as semi-solid flow batteries~\\cite{duduta2011semi}, it is critical to keep the contact between the active material (electrode particles) and the conductive wiring (carbon nanoparticle network) intact during operation, otherwise there is a significant energy loss during cycling of the battery~\\cite{solomon2018enhancing,wei2015biphasic}. This can be achieved by immersing the active material and the electronically conductive agent in yield-stress fluids like Carbopol~\\cite{zhu2020high}. Therefore, we can use the predicted $\\Delta P_c\/\\tau_y$ to determine the size of the active particles to optimize the design and operation of semi-solid electrodes.\n\nThe present model can also provide insights on the design of porous media. By taking advantage of the computational efficiency of the proposed network model, we can perform on-the-fly optimization to construct porous media with optimal mixing and transport properties~\\cite{lester2013chaotic,kirkegaard2020optimal}. Such ideas have recently been implemented in elastic networks with optimal phonon band structures~\\cite{ronellenfitsch2019inverse,ronellenfitsch2019chiral}, and we believe they can also be used for designing porous media immersed in yield-stress fluids.\n\nGiven their inherent node\/edge structure, network models are fairly simple to be analyzed using graph theoretical tools. The unique property of yield-stress fluid, namely the fluidization conditions, allows us to use algorithms that can find the minimum resistance pathways with minimal efforts. In coarse-grained domains, however, where the microscopic geometric irregularities are encoded in the heterogeneous `permeability' tensor~\\cite{hewitt2016obstructed}, graph theory tools might not be the most suitable ones. An alternative way to calculate the first open channel in a continuum with spatially variable properties is through methodologies used in physical chemistry to identify reaction pathways~\\cite{henkelman2000climbing,swenson2018openpathsampling}. There, the first open channel corresponds to the path that passes through the minimum energy barrier, namely the transition state point.\n\nThe present model considers only the case of viscoplastic materials and disregards the fluid elasticity~\\cite{saramito2007new,cheddadi2012steady,fraggedakis2016yielding1,fraggedakis2016yielding2} prior to yielding. Therefore, further analysis is required to connect the yield criterion to the elastic modulus of the fluid, which can provide insights for engineering both the porous medium and the fluid itself. Additionally, the yielding and\/or stoppage conditions might be further affected by possible thixotropic~\\cite{mewis2012colloidal} and kinematic hardening~\\cite{dimitriou2019canonical,gurtin2010mechanics} phenomena.\n\n\\section{\\label{sec:conc}Summary}\nIn this work, we presented a network model for yield-stress fluids in porous media that describe the effects of porosity $1-\\phi$ and microstructure properties in complex geometries. We demonstrated the capabilities of the model to predict the critical applied pressure drop required to open the first channel in the medium. Also, we compared our results to direct numerical simulations of the full fluid problem for Bingham fluids and we showed the accuracy and computational efficiency of the network-based models to solid-liquid transition. Finally, we discussed the implications of our model on the optimization and design of porous media.\n\n\\section*{Contributions}\nD.F. conceptualized, designed, and performed the analysis in the present study. E.C. and O.T. provided the fluid flow simulation results. D.F. wrote the manuscript. All authors contributed to the final manuscript. \n\n\\section*{Acknowledgment}\nD.F. (aka dfrag) wants to thank T. Zhou and M. Mirzadeh for insightful discussions related to the validity of the network model. The authors declare no competing interests.\n\n\\section*{\\label{sec:appendix}Appendix}\n\\subsection*{Porous medium and network generation}\nFor the generation of porous media that consist of non-overlapping disks, we implemented the random sequential addition (RSA) algorithm~\\cite{zhang2013precise,cule1999generating,torquato2006random}. The procedure described in~\\cite{torquato2006random} allows for the fast generation of randomly packed disks with the desired volume fractions. Due to the constraint of non-overlapping disks, all generated microstructures never exceed $\\phi=0.52$ in two dimensions. \n\nFor the generation of the network model, we implemented the maximal ball algorithm as described in~\\cite{silin2006pore,dong2009pore,al2007network}. The procedure allows us to get the radius $r_i$ and length $l_i$ for each edge. The maximal ball algorithm represents `fits' a circle\/sphere within each pore~\\cite{alim2017local}, its radius of which represents the $r_i$ we use in Eq.~\\ref{eq:least_res}. Other choices of $r_i$ can be considered (e.g. equivalent radius etc.), however, our choice for the local radius works well in predicting both the first open channel and the critical pressure drop required to open it. The generated network was represented by a graph with vertices $V$ and edges $E$ using the open source library NetworkX~\\cite{hagberg2008exploring}.\n\nDetails on the numerical simulation of the fluid flow problem shown in Fig.~\\ref{fig:validation}(a) can be found in~\\cite{chaparian2020complex}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVery recently IceCube Neutrino Observatory reported the detection of a high-energy muon neutrino event, IceCube-170922A of energy $\\sim 290$ TeV with a 56.5\\% probability of being a truly astrophysical neutrino \\cite{IceCube18a, IceCube18b}. The best-fit reconstructed arrival direction of the neutrino was consistent with the $0.1^0$ from the sky location of a flaring gamma ray blazar TXS 0506+056 \\cite{IceCube18a, Ansoldi18}. As a follow-up observation Fermi Large Area Telescope (LAT) Collaboration \\cite{Tanaka17} reported that the direction of origin of IceCube-170922A was consistent with the known gamma ray source TXS 0506+056 blazar which was in a state of enhanced emission with day scale variability \\cite{Keivani18} on 28 September 2017. The observed association of a high-energy neutrino with a blazar during a period of enhanced gamma-ray emission suggests that blazars may indeed be one of the long-sought sources of very-high-energy cosmic rays and hence these observations offers a unique possibility to explore the interrelate between energetic gamma rays, neutrinos, and cosmic rays.\n\nThe electromagnetic spectral energy distribution (SED) of the blazar TXS 0506+056 exhibits a double-hump structure which is a common feature of the non-thermal emission from blazars. The first hump, which peaking in the optical-ultraviolet range, is usually attributed to synchrotron radiation and the higher energy hump with peak energy in the GeV range, is often interpreted due to inverse Compton (IC) emission. An archival study of the time-dependent $\\gamma$-ray data over the last ten years or so reveal that the source was in quiescent stage most of the time, the flaring was noticed during the period July 2017 to September 2017. The averaged integrated flux above 0.1 GeV from TXS 0506+056 was found $(7.6 \\pm 0.2) \\times 10^{-8}$ $cm^{-2}s^{-1}$ during 2008 to 2017 from Fermi-LAT observations which in the week 4 to 11 July 2017 elevates to the level $(5.3 \\pm 0.6) \\times 10^{-7}$ $cm^{-2}s^{-1}$. The Astro-Rivelatore Gamma a Immagini Leggero (AGILE) gamma-ray telescope obtained flux of $(5.3 \\pm 2.1)\\times 10^{-7}$ $cm^{-2}s^{-1}$ during 10 to 23 September 2017. The Major Atmospheric Gamma Imaging Cherenkov (MAGIC) Telescopes detected a significant very-high-energy $\\gamma$-ray signal with observed energies up to about 400 GeV on 28 September 2017. Note that the Icecube observatory detected the neutrino event on 22 September 2017. It was found from optical to x-ray observations that the lower energy hump of the SED of the source does not show any noticeable time variation over the stated period of study.    \n\nSeveral efforts have been made so far to model the production of the detected neutrino event together with the electromagnetic (EM) observations from TXS 0506+056. Mainly two different production scenario, namely lepto-hadronic ($p\\gamma$) \\cite{Ansoldi18,Keivani18,Gao18,Cerruti19} and hadronic (pp) \\cite{Liu18,Sahakyan18} have been proposed in the literature to interpret the observations. A common feature of all the proposed models is that protons, like electrons, are also assumed to be accelerated to relativistic energies in the acceleration sites. Subsequently the accelerated protons interacting with low energy photons of blazar environment (lepto-hadronic interaction) and\/or with ambient matter produce high energy gamma rays and neutrinos.  \n\nAnsoldi et al. (2018) \\cite{Ansoldi18} showed that the measured neutrino event from the said blazar can be interpreted consistently with the EM observations by assuming a dense field of external low-energy photons originating outside of the jet as targets for photohadronic interactions. The lack of broadline signatures in the optical spectrum of TXS 0506+056 and other BL Lac objects suggest that such external photons emissions may be weak \\cite{Keivani18}. The model discussed in Ansoldi et al. (2018) \\cite{Ansoldi18}, however, does not invoke radiation from broadlines but, instead, assume the existence of soft radiation produced in a possible layer surrounding the jet. Therefore the lack of broadlines does not impact this specific scenario. In this context it is also to be noted that the BL Lac nature of TXS 0506+056 has been recently questioned by Padovani et al. 2019 \\cite{Padovani19}. Keivani et al. (2018) \\cite{Keivani18} considered a hybrid leptonic scenario of TXS 0506+056 where the production of high energy gamma rays was interpreted by external inverse-Compton processes and high-energy neutrinos via a radiatively sub-dominant hadronic component. \n\nFor efficient high energy $\\gamma-$ray production in AGN jet via $pp$ interaction demands high thermal plasma density; the thermal plasma in the jet should exceed $10^6$ cm$^{\u22123}$ in order to interpret the reported TeV flares of Markarian 501 by $pp$ interactions for any reasonable acceleration power of protons $L_p \\le 10^{45}$ erg\/s \\cite{Aharonian00}. The stated pure hadronic mechanism thus can be effectively realized in a scenario like ``relativistic jet meets target\" \\cite{Morrison84}, i.e. considering that $\\gamma-$radiation is produced in dense gas clouds that move across the jet \\cite{Dar97}. Recently, Liu et al. (2018) \\cite{Liu18} described the observed gamma ray \\& neutrino flux from the blazar TXS 0506+056 by assuming the presence of clouds in the vicinity of the super-massive black hole (SMBH) that provides targets for inelastic $pp$ collisions once they enter the jet. Liu et al. (2018) considered the synchrotron emission and inverse Compton emission of secondary electrons produced in cascade when high energy $\\gamma-$rays absorbed in $\\gamma\\gamma$ pair production with the emission region of the jet. However, the presence of broadline region (BLR) clouds in the vicinity of the SMBH for TXS 0506+056 is questionable due to the non-detection of the BLR emission from TXS 0506+056 and other BL Lac objects \\cite{Keivani18}. \n\nThe composition of bulk of the jet medium is not clearly known which makes difficulties to understand the interaction mechanism for gamma ray and neutrino production. But on average, jet plasma must be neutral to remain collimated \\cite{Hirotani05}. Therefore, the two main scenarios for their matter composition are suggested: a `pair plasma' consisting of only of relativistic electrons and positrons \\cite{Kino04} and a `normal plasma' consisting of (relativistic or non-relativistic) protons and relativistic electrons \\cite{Celotti93}. A useful quantity that can furnish some constraints on jet composition is the kinetic power of an AGN jet. By comparing the bulk kinetic energy of the parsec scale jet with the kinetic luminosities on extended scales \\cite{Rawlings91}, Celotti \\& Fabian (1993) \\cite{Celotti93} argued in favor of an electron-proton fluid. For high luminous blazars, to maintain the radiated power which would not exceed that carried by the jet, the proton component of plasma is necessary (see Ghisellini, 2010 \\cite{Ghisellini10}, and references therein). \n\nUnder the context, in the present work we exploit the main essences of proton blazar model \\cite{Protheroe01, Mucke01} to explain the observed higher-energy bump of the EM SED along with the neutrino from the blazar TXS 0506+056 at flaring stage. The detected lower-energy bump of EM SED from the blazar can be well interpreted with the synchrotron radiation of relativistic electrons present in jet plasma whereas the cold (non-relativistic) protons density that arose from charge neutrality condition can provide sufficient target matter (proton) for production of high energy gamma rays and neutrinos via the $pp$ interaction. For TXS 0506+056 such a scenario is more realistic then the scenario like the cloud-in-jet model \\cite{Aharonian17} as we argue later. We would also like to examine the maximum energy that a cosmic ray particle can attain in the blazar jet; the detected $\\sim 290$ TeV energy neutrino alone suggest that acceleration of protons in the jet of this object to energies of at least several times $10^{15}$ eV. \n\nThe organization of this paper is as follows: In the next section, we shall describe the methodology for evaluating the gamma-ray and neutrino  fluxes generated in interaction of cosmic rays with the ambient matter in the AGN jet under the framework of proton blazar model. The numerical results of the hadronically produced gamma-rays and neutrino fluxes from the AGN jet over the GeV to TeV energy range are shown in section III. The findings are compared with the observed gamma rays spectra and the neutrino event from the blazar and the results are discussed in the same section. Finally we conclude in Sec. IV.\n\n\\section{Methodology}\n\nThe overall jet composition of AGN is not properly known. In the adopted proton blazar inspired model it is assumed that the relativistic jet material is composed of relativistic protons (p) and electrons (e$^\u2212$). Some cold protons also exist, allowing charge neutrality to be fulfilled. The ratio of number of relativistic protons to electrons, the maximum energies attained by protons\/electrons in acceleration process and slope of their energy spectrum, luminosities of electrons and protons are adjustable parameters of the model. In this model flaring is produced due to high magnetic activities in the source (similar to the origin of flaring activities in the Sun). \n\nWe consider a spherical blob of size $R_b'$ (primed variables for jet frame) in the AGN jet which is the region responsible for the blazar emission. The blob is moving with a Doppler factor $\\delta = \\Gamma_j^{-1}(1-\\beta_j\\cos\\theta)^{-1}$ where $\\theta$ is the angle between the line of sight and the jet axis and $\\Gamma_j = 1\/\\sqrt{1-\\beta_j^2}$ is the bulk Lorentz factor \\cite{Petropoulou15}, and it contains a tangled magnetic field of strength $B'$.  \n\nIn the proton blazar framework the low-energy bump of the SED is explained by synchrotron radiation of accelerated relativistic electron in blazar jet having broken power law energy distribution as \\cite{Katarzynski01}\n\n\\begin{eqnarray}\nN_e'(\\gamma_e') = K_e \\gamma_e'^{-\\alpha_1} \\hspace{1.5cm} \\mbox{if}\\hspace{0.6cm} \\gamma_{e,min}' \\le \\gamma_e' \\le \\gamma_b' \\nonumber \\\\\n         = K_e \\gamma_b'^{\\alpha_2-\\alpha_1} \\gamma_e'^{-\\alpha_2} \\hspace{0.35cm} \\mbox{if}\\hspace{0.45cm} \\gamma_b' <\\gamma_e' \\le \\gamma_{e,max}'\\;\n\\end{eqnarray}\nwhere $\\gamma_e' = E_e'\/m_e c^2$ is  the  Lorentz  factor of electrons of energy $E_e'$, $\\alpha_1$ and $\\alpha_2$ are the spectral indices before and after the spectral break Lorentz factor $\\gamma_b'$ respectively. The normalization constant $k_e$ can be found from \\cite{Bottcher13}\n\\begin{equation}\nL_e' = \\pi R_b'^2 \\beta_j c \\int_{\\gamma_{e,min}'}^{\\gamma'_{e,max}}m_e c^2\\gamma_e' N_e(\\gamma_e') d\\gamma_e'\n\\end{equation}\nwhere $L_e'$ is the kinetic power in relativistic electrons in the blazar jet frame. The number density of highly relativistic (`hot') electrons is $n_{e,h}' = \\int N_e'(\\gamma_e') d\\gamma_e'$ and the corresponding energy density is $u_e' = 3p_e' = \\int m_e c^2\\gamma_e' N_e'(\\gamma_e') d\\gamma_e'$ where $p_e'$ is the radiation pressure due to relativistic electrons. Due to strong synchrotron and Inverse Compton cooling at relativistic energies, the acceleration efficiency of electrons in AGN jet is quite low and it can be assumed to be $\\chi_e \\approx 10^{-3}$ \\cite{Bykov96,Eichler05,Vazza15}. Hence total number can be determined as $n_{e}' = n_{e,h}'\/\\chi_e$. Thus the number density of non-relativistic (`cold') electrons is given by $n_{e,c}' = n_{e}' - n_{e,h}'$.\n\nThe emissivity of photons of energy $E_{s}'$ ($= m_e c^2 \\epsilon_{s}'$) due to the synchrotron emission of electrons which describe low energy component of EM SED of the blazar, can be written as \\cite{Bottcher13}\n\\begin{eqnarray}\nQ'_{s}(\\epsilon_{s}') = A_0 \\epsilon_{s}'^{-3\/2} \\int_1^\\infty d\\gamma_e' N_e'(\\gamma_e')\\gamma_e'^{-2\/3}e^{-\\epsilon_{s}'\/(b\\gamma_e'^2)}\n\\end{eqnarray}\nwith the normalization constant\n\\begin{eqnarray}\nA_0 = \\frac{c \\sigma_T B'^2}{6\\pi m_e c^2 \\Gamma(4\/3) b^{4\/3}}, \\nonumber\n\\end{eqnarray}\nwhere $\\sigma_T$ is the Thomson cross-section, $b = B'\/B_{crit}$ and $B_{crit} = 4.4\\times 10^{13}$ G. The magnetic field energy density is $u_B' = B'^2\/8\\pi = 3 p_B'$ where $p_B'$ is the corresponding pressure.\n\nThe emissivity of photons of energy $E_{c}'$ ($= m_e c^2 \\epsilon_{c}'$) due to the inverse compton scattering of primary accelerated electrons\nwith the seed photons co-moving with the AGN jet, which can describe lower part of high energy component of EM SED of the blazar, can be written as\\cite{Blumenthal70,Inoue96}\n\n\\begin{eqnarray}\nQ_{c}(\\epsilon_{c}') = \\int_0^\\infty d\\epsilon_{j}' n_j'(\\epsilon_{j}') \\int_{\\gamma_{e,0}'}^{\\gamma_{e,max}'} d\\gamma_e' N_e'(\\gamma_e') C(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}'),\n\\end{eqnarray}\nwhere $\\gamma_{e,0}' = \\frac{1}{2}\\epsilon_{c}'\\left(1+\\sqrt{1+\\frac{1}{\\epsilon_{c}'\\epsilon_{j}'}}\\right)$ and the compton kernel $C(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}')$ is given by Jones (1968) \\cite{Jones68} as\n\\begin{eqnarray}\nC(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}') = \\frac{2\\pi r_e^2 c}{\\gamma_e'^2\\epsilon_{j}'} \\Big[2k \\ln(k) + (1+2k)(1-k)   \\big.  \\nonumber   \\\\\n\\big. + \\frac{(4\\epsilon_{j}'\\gamma_e' k)^2}{2(1+4\\epsilon_{j}'\\gamma_e'k)}(1-k)\\Big],\n\\end{eqnarray}\nwith $k = \\frac{\\epsilon_{c}'}{4\\epsilon_{j}'\\gamma_e'(\\gamma_e' - \\epsilon_{c}')}$ and $r_e$ is the classical electron radius. Here, $n_{j}'(\\epsilon_{j}')$ is the average number density of the seed photons of energy $\\epsilon_{j}'$( in $m_ec^2$) in the blob of AGN jet which can be directly related to observed photon flux $f_{\\epsilon_j}$ (in erg cm$^{-2}$ s$^{-1}$) from the blazar through \\cite{Dermer02}\n\\begin{eqnarray}\n\\epsilon_{j}'n_{j}'(\\epsilon_{j}') = \\frac{2 d_L^2}{c R_b'^2\\delta^2\\Gamma_j^2} \\frac{f_{\\epsilon_j}}{m_e c^2\\epsilon_{j}' }\n\\end{eqnarray}\nwhere $\\epsilon_{j} = \\delta \\epsilon_{j}'\/(1+z) $ \\cite{Atoyan03} relates photon energies in the observer and co-moving jet frame of red shift parameter $z$ respectively,  and $d_L$ is the luminosity distance of the AGN from the Earth. \n\nIn the proton blazar model the cosmic ray protons are also supposed to accelerate to very high energies $E_p' = m_p c^2 \\gamma_p'$ in the same region of blazar jet and the production  spectrum shall follow a power law \\cite{Malkov01,Cerruti15}:  \n\\begin{equation}\n N_p'(\\gamma'_p) =  K_p {\\gamma'_p}^{-\\alpha_p} .\n\\end{equation}\nwhere $\\alpha_p$ is the spectral index, $\\gamma_p'$ is the Lorentz factor of accelerated protons, $K_p$ denotes the proportionality constant which can be found from the same expression as eq (2) but for protons and $L_p'$ is the corresponding jet power in relativistic protons. The number density of relativistic protons is $n_p' = \\int N_p'(\\gamma_p') d\\gamma_p' $ and the corresponding energy density is $u_p' = 3p_p' = \\int m_p c^2\\gamma_p' N_p'(\\gamma_p') d\\gamma_p'$, where $p_p'$ is the radiation pressure due to relativistic protons. \n\nWe estimate the mechanical luminosity or total kinematic jet power of an AGN jet containing jet frame energy density $u'$ (sum of $u_e'$, $u_p'$ and $u_B'$), pressure $p'$ (sum of $p_e'$, $p_p'$ and $p_B'$) and matter density $\\rho'$ (including cold protons and electrons) from the following relation \\cite{Protheroe01}\n\\begin{eqnarray}\nL_{jet} = \\Gamma_j^2 \\beta_j c \\pi R_b'^2\\left[\\rho'c^2(\\Gamma_j-1)\/\\Gamma_j + u' + p' \\right].\n\\end{eqnarray}\nwhere we assume the Lorentz factor to be $\\Gamma \\approx \\delta\/2$ which is quite reasonable particularly for jets closely aligned to the line of sight of the observer. Applying charge conservation and considering that the number of relativistic electrons will be greater then the number of relativistic protons, the number of `cold' (non-relativistic) protons will be equal to the total number of electrons ($n_{e}'$) minus the number of hot protons ($n_p'$). Thus the cold matter density in protons and in electrons in the blob will be $\\rho_p' = (n_{e}'-n_p')m_p$ where $m_p$ is the rest mass of a proton and $\\rho_e' = n_{e,c}' m_e$ respectively. \n\nWhen the shock accelerated cosmic rays interact with the cold matter (protons) of density $n_{H} = \\rho_p'\/ m_p$ in the blob of AGN jet, the emissivity of produced secondary particles of energy $E_i' = m_e c^2 \\epsilon_i'$ in co-moving AGN jet frame is given by \\cite{Liu18,Anchordoqui07,Banik17a,Kelner06}\n\\begin{eqnarray}\nQ_{i,pp}'(\\epsilon'_{i}) = \\frac{c n_{H}m_e}{m_p} \\int_{\\frac{m_e\\epsilon'_{i}}{m_p}} \\sigma_{pp}(E'_{p}) N'_p(\\gamma'_{p})F_i\\Big(\\frac{E'_{i}}{E'_{p}},E'_{p}\\Big)\\frac{d\\gamma'_{p}}{\\gamma'_{p}}\n\\end{eqnarray}\nwhere $i$ could be $\\pi^{0}$ mesons, electrons (positrons) $e^{\\pm}$ or neutrinos $\\nu$ and $F_i$ is the spectrum of the corresponding secondary particles in a single $pp$ collision as given in Kelner et al. (2006) \\cite{Kelner06}.\n\nDue to decay of $\\pi^0$ mesons, the resulting gamma ray emissivity as a function of gamma ray energy $E_{\\gamma}'( = m_e c^2 \\epsilon_{\\gamma}')$ is given by \\cite{Banik17b} \n\n\\begin{eqnarray}\nQ_{\\gamma,pp}'(\\epsilon'_{\\gamma}) = 2\\int_{\\epsilon_{\\pi,min}'(\\epsilon'_{\\gamma})}^{\\epsilon_{\\pi,max}'}\\frac{Q_{\\pi,pp}'(\\epsilon'_{\\pi})}{\\left[{\\epsilon'_{\\pi}}^2-(\\frac{m_{\\pi}}{m_e})^2\\right]^{1\/2}}d\\epsilon'_{\\pi}\n\\end{eqnarray}\nwhere ${\\epsilon'_{\\pi,min}}(\\epsilon'_{\\gamma}) = \\epsilon'_{\\gamma} + (\\frac{m_{\\pi}}{m_e})^2\/(4\\epsilon'_{\\gamma})$ is the minimum energy of a pion required to produce a gamma ray photon of energy $\\epsilon'_{\\gamma}$ (in $m_e c^2$).\n\nWhen propagating through an isotropic source of low-frequency radiation, the TeV$-$PeV gamma-rays can be absorbed at photon-photon ($\\gamma\\gamma$) interactions \\cite{Aharonian08}. Thus, the emissivity of escaped gamma rays after $\\gamma\\gamma$-interaction can be written as \\cite{Bottcher13} \n\\begin{eqnarray}\nQ_{\\gamma,esc}'(\\epsilon_{\\gamma}') = Q_{\\gamma}'(\\epsilon_{\\gamma}') .\\left( \\frac{1-e^{-\\tau_{\\gamma \\gamma}}}{\\tau_{\\gamma \\gamma}} \\right).\n\\end{eqnarray}\nHere $\\tau_{\\gamma \\gamma}(\\epsilon_{\\gamma}')$ is the optical depth for the interaction and is given by \\cite{Aharonian08}\n\\begin{eqnarray}\n\\tau_{\\gamma \\gamma}(\\epsilon_{\\gamma}') = R_b' \\int \\sigma_{\\gamma\\gamma}(\\epsilon_{\\gamma}',\\epsilon_{j}') n_{j}'(\\epsilon_{j}')d\\epsilon_{j}'\n\\end{eqnarray}\nwhere $\\sigma_{\\gamma\\gamma}$ is the the total cross-section as given in Aharonian et al., 2008 \\cite{Aharonian08} and $n_{j}'(\\epsilon_{j}')$ describes the spectral distributions of target photons.  $n_{j}'(\\epsilon_{j}')$ is generally assumed to be the observed synchrotron radiation photons produced by the relativistic electron population in co-moving jet frame as given in eq.(6) because of the low luminosity of accretion disks in BL Lacs \\cite{Mucke01}.\n\nThe number of injected electrons (positrons) per unit volume and time in AGN blob with a Lorentz factor $\\gamma_e'$ coming from $\\gamma \\gamma$ pair production of high-energy photons as given by Aharonian,Atoian \\& Nagapetian (1983) \\cite{Aharonian83} reads\n\n\\begin{eqnarray}\nQ_{e,\\gamma\\gamma}'(\\gamma_e') = \\frac{3 \\sigma_T c}{32} \\int_{\\gamma_e'}^\\infty d\\epsilon_{\\gamma}' \\frac{n'_{\\gamma}(\\epsilon_{\\gamma}')}{\\epsilon_{\\gamma}'^3} \\int_{\\frac{\\epsilon_{\\gamma}'}{4\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}}^\\infty d\\epsilon_{j}' \\frac{n_{j}'(\\epsilon_{j}')}{\\epsilon_{j}'^2}  \\nonumber  \\\\\n\\times \\left[ \\frac{4\\epsilon_{\\gamma}'^2}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}\\ln\\left( \\frac{4\\gamma_e'\\epsilon_{\\gamma}'(\\epsilon_{\\gamma}'-\\gamma_e')}{\\epsilon_{\\gamma}'}\\right) - 8\\epsilon_{\\gamma}'\\epsilon_{j}' \\right.  \\nonumber   \\\\\n\\left. + \\frac{2\\epsilon_{\\gamma}'^2(\\epsilon_{\\gamma}'\\epsilon_{j}'-1)}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}-\\left( 1-\\frac{1}{\\epsilon_{\\gamma}'\\epsilon_{j}'}\\right)\\left(\\frac{\\epsilon_{\\gamma}'^2}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}\\right)^2   \\right]  \\;\n\\end{eqnarray}\nwhere $n'_{\\gamma}(\\epsilon'_{\\gamma}) = (R_b'\/c)Q_{\\gamma,pp}'$ is the number density of photons of high energy $\\epsilon_{\\gamma}'$.\n\nThe high-energy injected electrons\/positrons ($Q_{e}'$) including both those ($Q_{e,\\gamma\\gamma}'$), produced in $\\gamma\\gamma$ pair production and those ($Q_{e,\\pi}'$), created directly due to the decay of $\\pi^{\\pm}$ mesons produced in $pp$ interaction (using eq.(9)) will initiate EM cascades in the AGN blob via the synchrotron radiation, the IC scattering.\n\nIn order to determine the stationary state of the population of produced electron distribution $N_e'(\\gamma_e')$, the injection function $Q_e'(\\gamma_e')$ has been used as a source term in the continuity equation for electrons as given by \\cite{Cerruti15}\n\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t} \\left[N_e'(\\gamma_e')\\right] = \\frac{\\partial}{\\partial \\gamma_e'}\\left[\\gamma_e'\\frac{N_e'(\\gamma_e')}{\\tau_c(\\gamma_e')}\\right] + Q_e'(\\gamma_e') - \\frac{N_e'(\\gamma_e')}{\\tau_{ad}},\n\\end{eqnarray}\nwhere we consider the adiabatic time-scale as $\\tau_{ad} = 2R_b'\/c$. The radiative cooling time, considering both inverse Compton losses and synchrotron losses is given by \\cite{Cerruti15}\n\n\\begin{eqnarray}\n\\tau_c(\\gamma_e') = \\frac{3m_e c}{4(u_B' + u_{ph}')\\sigma_T}\\frac{1}{\\gamma_e'}\n\\end{eqnarray}\nwhere $u_{ph}'$ is the energy density of photons in co-moving jet frame in equilibrium. \n\nUsing the integral expression given by Inoue \\& Takahara (1996) \\cite{Inoue96}, the solution of eq. (14) i.e, the cascade electron distribution in stationary state can be evaluated as  \n\n\\begin{eqnarray}\nN_e'(\\gamma_e') = e^{-\\gamma_e^*\/\\gamma_e'}\\frac{\\gamma_e^*\\tau_{ad}}{\\gamma_e'^2}\\int_{\\gamma_e'}^\\infty d\\zeta Q_e'(\\zeta)e^{+\\gamma_e^*\/\\zeta}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma_e^* = \\frac{3m_e c^2}{8(u_B' + u_{ph}')\\sigma_T R_b'}\n\\end{eqnarray}\nindicating the Lorentz factor of electron when $\\tau_c(\\gamma_e') = \\tau_{ad}$. Once the equilibrium pair distribution $N_e'(\\gamma_e')$ is known, the associated stationary synchrotron emission is evaluated using Equation (3) and hence found the observable photon spectrum using eq. (11). \n\nLet $Q_{\\gamma,esc}'(\\epsilon_{\\gamma}')$ be the total gamma ray emissivity from the blob of AGN jet including all processes stated above i.e, the synchrotron and the IC radiation of relativistic electrons, the gamma rays produced in $pp$ interaction and also the synchrotron photons of EM cascade electrons. The observable differential flux of gamma rays reaching at the earth, therefore, can be written as   \n\n\\begin{eqnarray}\nE_{\\gamma}^2\\frac{d\\Phi_{\\gamma}}{dE_{\\gamma}} = \\frac{V'\\delta^2\\Gamma_j^2}{4\\pi d_L^{2}}\\frac{E_{\\gamma}'^2}{m_e c^2} Q_{\\gamma,esc}'(\\epsilon_{\\gamma}') . e^{-\\tau_{\\gamma\\gamma}^{EBL}}\n\\end{eqnarray}\nwhere $E_{\\gamma} = \\delta E_{\\gamma}'\/(1+z) $ \\cite{Atoyan03} relates photon energies in the observer and co-moving jet frame of red shift parameter $z$ respectively with $E_{\\gamma}' = m_e c^2 \\epsilon_{\\gamma}'$, $V' = \\frac{4}{3}\\pi R_b'^3$ is the volume of the emission region. Here we employ the Franceschini-Rodighiero-Vaccari (FRV) model \\cite{Franceschini08,website1} to find the optical depth $\\tau_{\\gamma\\gamma}^{EBL}(\\epsilon_{\\gamma}',z)$ for gamma-ray photons due to the absorption by the extragalactic background (EBL) light.\n\nThe corresponding flux of muon neutrinos reaching at the earth can be written as \n\\begin{eqnarray}\nE_{\\nu}^2\\frac{d\\Phi_{\\nu_{\\mu}}}{dE_{\\nu}} = \\xi.\\frac{V'\\delta^2\\Gamma_j^2}{4\\pi d_L^{2}} \\frac{E_{\\nu}'^2}{m_e c^2}Q_{\\nu,pp}'(\\epsilon_{\\nu}') \n\\end{eqnarray}\nwhere $E_{\\nu} = \\delta E_{\\nu}'\/(1+z) $ \\cite{Atoyan03} relates neutrino energies in the observer and co-moving jet frame respectively and the fraction $\\xi = 1\/3$ is considered due to neutrino oscillation.\n\n\\section{Numerical results and discussion}\nIn the third catalog of AGNs detected by Fermi-LAT listing 1773 objects \\cite{Ackermann15}, TXS 0506+056 is one of the most luminous objects with an average flux of $6.5 (\\pm 0.2)\\times 10^{-9}$ photons cm$^{-2}$ s$^{-1}$ between 1 GeV and 100 GeV. A high-energy neutrino-induced muon track IceCube-170922A, detected on 22 September 2017, was found to be positionally coincident with the flaring $\\gamma-$ray blazar, TXS 0506+056 \\cite{IceCube18a}. The coincidence detection probability by chance was found to be disfavored at a $3\\sigma$ confidence level mainly due to the precise determination of the direction of neutrino \\cite{IceCube18a} although no additional excess of neutrinos was found from the direction of TXS 0506+056 near the time of the alert. Assuming a spectral index of \u22122.13 (\u22122.0) for the diffuse astrophysical muon neutrino spectrum \\cite{Aartsen14}, the most probable energy of the neutrino event was found to be 290 TeV (311 TeV) with the 90\\% C.L. lower and upper limits being 183 TeV (200 TeV) and 4.3 PeV (7.5 PeV), respectively \\cite{IceCube18a, Ansoldi18}. \nExtensive follow-up observations by the Fermi-Large Area Telescope \\cite{Tanaka17} in GeV gamma-rays and by the Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC) \\cite{Mirzoyan17} telescopes in very-high-energy (VHE) gamma-rays above 100 GeV, revealed TXS 0506+056 to be active in all EM bands. The redshift of the blazar has been recently measured to be $z = 0.3365$ \\cite{Paiano18} and the luminosity distance, estimated with a consensus cosmology is $d_L \\sim 1750$ Mpc \\cite{Keivani18}.\n\nThe gamma ray variability time scale is found as $t_{ver} \\le 10^5$ s by analyzing the X-ray and gamma-ray light curves \\cite{Keivani18}. Consequently to describe the electromagnetic SED of TXS 0506+056 over the optical to gamma ray energy range we have chosen the size of emission region of $R_b' = 2.2\\times10^{16}$ cm with Doppler boosting factor $\\delta = 20$ and bulk Lorentz factor of AGN jet $\\Gamma_j = 10.4$  which are strongly consistent with the size inferred from the variability, namely $R_b' \\lesssim \\delta c t_{ver}\/(1+z) \\simeq 4.5\\times10^{16} (\\delta\/20) (t_{ver}\/10^5 s)$ cm \\cite{Keivani18}. \n\nThe low energy part of the experimental EM SED data can be explained well by synchrotron emission of primary relativistic electron's distribution obeying a broken power law as given by Eq.(1) with spectral indices $\\alpha_1 = 1.71$ and $\\alpha_2 = 4.3$ respectively before and after the spectral break Lorentz factor $\\gamma_b' = 8.5\\times 10^{3}$. The required kinematic power of relativistic electrons in blazar jet as given by Eq. (2) and the magnetic field to fit the observed data are $L_e' = 2.3\\times 10^{42}$ erg\/s and $B' = 0.38$ G respectively. Here we have not included the self-absorption of synchrotron photons spectrum. When the self-absorption mechanism \\cite{Katarzynski01} is included, the resultant spectrum will show slight mismatch with the observed photon flux at radio energies, particularly VLA and OVRO data.\n\nThe inverse Compton scattering of primary electrons with the target synchrotron photons (also including high energy photons) co-moving with the AGN jet as given by Eq. (4) are also found to produce lower part of high energy bump of EM spectrum, particularly from NuSTAR experimental data upto Fermi-LAT data.  The number of `hot' electrons in blob of the AGN jet are estimated to be $n_{e,h}' = 1.7\\times10^3$ particles\/cm$^{3}$ which is required to produce the EM SED due to both synchrotron and inverse Compton emission. But the acceleration efficiency of electrons in AGN jet may be quite low and it can be assumed to be $\\chi_e \\approx 10^{-3}$ \\cite{Bykov96,Eichler05,Vazza15} due to strong synchrotron and Inverse Compton cooling at relativistic energies and the total number of electrons including `cold' electrons are found out to be $n_e = 1.7\\times10^6$ particles\/cm$^{3}$. \n\nIn the original proton blazar model high energy gamma rays are produced through synchrotron radiation by high energy protons in strong magnetic field environment. However, due to low magnetic field strength of the source (obtained from the fitting of low energy hump of SED) the gamma ray spectrum of the source can not be modeled with the proton synchrotron radiation. The proton-photon interaction is also found inefficient in the present case  due to low amplitude of target synchrotron photon field. Instead required gamma rays are found to produce in interactions of relativistic protons with the ambient cold protons in the blob. The observed higher energy part of observed EM SED data, particularly those measured with Fermi-LAT and MAGIC observatory, can be reproduced well by the model as estimated following the best fit Eq. (18). The spectral index of the energy spectrum of AGN accelerated cosmic rays is taken as $\\alpha_p = - 2.13$ which is consistent with the best fit spectral slope of the observed astrophysical neutrinos of between 194 TeV and 7.8 PeV by IceCube observatory \\cite{Halzen17,Aartsen16}. The required accelerated primary proton injection luminosity is found to be $L_p' = 10^{46}$ erg\/s. The cold proton number density in jet turns out to be $1.68\\times 10^6$ particles\/cm$^3$ under charge neutrality condition which provides sufficient targets for hadronuclear interactions with accelerated relativistic protons. The estimated differential gamma-ray spectrum reaching at Earth from this AGN is shown in Fig. 1 along with the different space and ground based observations. It is clear from the figure that the observed spectrum is correctly reproduced by the model. The detection sensitivity of upcoming gamma-ray experiments like the Cherenkov telescope array (CTA) \\cite{Ong17} and the Large High Altitude Air Shower Observatory (LHAASO) \\cite{Liu17} are also shown in the figure which suggest that these experiments will be able to detect gamma rays up to nearly 100 TeV for any similar kind of event if detected in future and thereby will be able to provide a better understanding of the emission processes.\n\n\\begin{figure}[h]\n  \\begin{center}\n  \\includegraphics[width = 0.5\\textwidth,height = 0.45\\textwidth,angle=0]{txs_report_final_16_new.eps}\n\\end{center}\n\\caption{Estimated differential energy spectrum of gamma rays and neutrinos reaching at the Earth from the blazar TXS 0506+056. The pink small dashed line indicates the low energy component of EM spectrum due to synchrotron emission of relativistic electrons. The green long dash-double-dotted denotes the gamma ray flux produced due to Inverse Compton emission of relativistic electrons in seed photon distribution in co-moving jet frame. The red dotted line represents the gamma ray flux produced from neutral pion decay in $pp$-interaction together with the cascade emission of electron\/positron produced in (pionic) $\\gamma\\gamma$-absorption. The black continuous line represents the estimated overall differential multi wave-length EM SED. The blue small dash-single dotted line indicates the differential muon neutrino flux reaching at earth. The yellow dash-triple-dotted line and brown long dash-single dotted line denote the detection sensitivity of the CTA detector for 1000 h and the LHAASO detector for 1 year respectively. The cyan long dashed line indicates the expected level\\cite{Gao18} and energy range of the neutrino flux reaching at earth  to produce one muon neutrino in IceCube in 0.5 year, as observed.}  \n\\label{Fig:1}\n\\end{figure}\n\nHigh energy neutrinos are produced together with gamma rays in pp interactions. The high energy neutrino flux at the Earth from the blazar TXS 0506+056 in active state has also been estimated following Eq. (19) and also shown in Fig. 1 along with the concerned Icecube result. For estimation of the neutrino flux no additional adjustable parameters were available; the same parameters used to describe gamma ray spectrum lead the neutrino flux. The total mechanical jet power of the blazar in jet frame is found out to be $L_{jet}' = 1.2\\times 10^{47}$ erg\/s and physical jet power after Lorentz boost is $L_{jet} = 1.3\\times 10^{49}$ erg\/s. It is noticed that $\\eta'_p = L_p'\/L_{jet} = 8.5\\%$ under assumption of electron injection efficiency about $\\chi_e \\approx 10^{-3}$ i.e, cosmic ray protons carries 8.5\\% energy of total jet power in co-moving jet frame which is generally expected acceleration efficiency of cosmic rays in astrophysical sources \\cite{Banik17b,Sahakyan18}. \nThe total jet power in the form of magnetic field and relativistic electron and proton kinetic energy calculated as \\cite{Ansoldi18} $L^k_{jet} = \\Gamma_j^2 \\beta_j c \\pi R_b'^2\\left[u_e' + u_p' + u_B' \\right]$ and found out to be $10^{48}$ erg\/s. The estimated kinetic jet power of the blazar is consistent with the Eddington luminosity of $L_{edd}\\gtrsim 1.3\\times 10^{48}$ erg\/s if we assume a super-massive black hole of mass $M_{bh} \\gtrsim 10^{10} M_{\\odot}$, like blazar S5 $0014+813$ \\cite{Ghisellini09}. However, jet power may exceed the Eddington luminosity during outbursts or for a collimated outflow in a jet because in such situations the jet does not interfere with the accretion flow. Note that a moderate excess of jet power over the Eddington luminosity (within a factor of ten) seems physically viable \\cite{Gao18,Sadowski15}.\n\nThe MAGIC collaboration reported the prominent spectral steepening observed gamma ray spectra from the said blazar above $\\sim 100$ GeV which confirms the internal $\\gamma\\gamma$ absorption that is robustly expected as a consequence of $pp$ production of a $\\sim 290$ TeV neutrino and also restrict the $\\delta$ to a low value. The cascade emission of electron\/positron pairs induced by protons has been estimated following Eq. ($14-16$) where we include the contribution of high energy photons along with synchrotron photons in jet frame as target for internal $\\gamma\\gamma$ absorption of firstly produced gamma rays in $pp$ interaction. This mechanism also found to contribute significantly in the hard X-ray to VHE gamma-ray bands. A primary cosmic ray proton spectrum up to $E_{p,max}' = 10$ PeV in jet frame and magnetic fields of $B' = 0.38$ G which is mutually consistent with the synchrotron radiation of electrons for lower bump in EM SED, can somewhat describe the observed gamma-ray spectra. \n\nThe number of expected muon neutrino event in time $\\tau$ can be found from the relation $N_{\\nu_{\\mu}} = \\tau \\int_{\\epsilon_{\\nu,min}}^{\\epsilon_{\\nu,max}} A_{eff}(\\epsilon_{\\nu}). \\frac{d\\phi_{\\nu_{\\mu}}}{d\\epsilon_{\\nu}} d\\epsilon_{\\nu}$ where $A_{eff}$ be the IceCube detector effective area at the declination of the TXS 0506+056 in the sky \\cite{IceCube18b,Padovani18,Albert18}. We found that the expected muon neutrino event in IceCube detector from the blazar in 200 TeV and 7.5 PeV energy range are about $N_{\\nu_{\\mu}} = 1.007$ events in 0.5 years for the flaring VHE emission state with $E_{p,max}' = 10$ PeV. The expected muon neutrino event are about $N_{\\nu_{\\mu}} = 2.6$ for the same scenario but in the energy range of 32 TeV and 3.6 PeV with $E_{p,max}' = 10$ PeV which is in good agreement with the effective energy range \\cite{IceCube18b} of IceCube for astrophysical neutrinos. The model fitting parameters to match the EM SED as well as muon neutrino event are summarized in Table~\\ref{table1}.\n\n\\begin{table}[h]\n  \\begin{center}\n    \\caption{Model fitting parameters for TXS 0506+056 according to proton blazar model.}\n    \\label{table1}\n    \\begin{tabular}{c|c}\n      \\toprule\n      Parameters &  Values   \\\\ \\hline\n      $\\delta$              &  $20$ \\\\\n      $\\Gamma_j$              &   $10.4$  \\\\\n      $\\theta$              &   $1^{0}$  \\\\\n       $z$                  &  $0.3365$   \\\\\n      $R_b'$  (in cm)            &  $2.2\\times 10^{16}$ \\\\\n       $B$ (in G)          &  $0.38$  \\\\\n       $u_B$ (in erg\/cm$^{3}$)          &  $5.75\\times10^{-3}$  \\\\\n       $\\alpha_1$           &   $- 1.71$  \\\\ \n       $\\alpha_2$           &   $- 4.3$  \\\\\n       $\\gamma_b'$  &  $8.5\\times 10^{3}$ \\\\\n       $\\gamma_{e,min}'$  &  $1$ \\\\\n       $\\gamma_{e,max}'$  &  $1.5\\times 10^{5}$ \\\\\n       $u_e$ (in erg\/cm$^{3}$)          &  $4.5\\times10^{-2}$  \\\\\n       $L_e'$ (in erg\/s)    & $2.3\\times 10^{42}$  \\\\ \n       $n_H$  (in cm$^{-3}$) &  $1.68\\times 10^6$  \\\\\n       $\\alpha_p$           &   $- 2.13$  \\\\\n       $E_{p,max}'$ (in eV) &  $ 10^{16}$  \\\\ \n       $L_p'$ (in erg\/s)    &  $10^{46}$  \\\\\n       $L^k_{jet}$ (in erg\/s)   &  $ 10^{48}$ \\\\\n       $N_{\\nu_{\\mu}}$   &  1.007  \\\\ \\hline    \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\nThe VHE gamma-ray flux from the blazar is found to be variable i.e, increasing by a factor of up to $\\sim 6$ within one day from low state (quiescent state) to the flaring state. The flux variability found mainly in high energy component but not in lower bump of EM spectra from the source disfavors the inverse-Compton origin for such variabilities. There may be two possible scenarios for such variabilities$-$ i) The VHE gamma-ray flux in low state is leptonic in origin, i.e, via inverse-Compton emission from electrons up-scattering synchrotron photons (synchrotron-self-Compton scenario, SSC \\cite{Maraschi92,Bloom96,Mastichiadis97}) or photons from the ambient fields (external inverse-Compton, EIC \\cite{Dermer92,Dermer93}) but consequently no neutrinos will produce. The higher flux of gamma rays in flaring state can be interpreted when the blazar jet meets with the external cloud \\cite{Aharonian17,Barkov12,Dar97} which will provide sufficient target matter (protons) for interaction with accelerated cosmic rays to produce observed high energy gamma rays and neutrinos efficiently. ii) The VHE gamma-ray flux in both low state and flaring state can be explained in a hadronic interaction model using a proton blazar model. In this scenario, observed gamma ray flux can be explained well with hadronic $pp$ interaction of accelerated cosmic rays of comparatively harder spectral slope ($\\sim 2.28$) and lowering the maximum energy of accelerated cosmic rays with ambient `cold' proton (in charge neutrality condition with co-accelerated electrons) in low state of the blazar compared to flaring state and subsequently produce neutrinos (of event $N_{\\nu_{\\mu}} = 0.13$ in 0.5 year) as well.  \n\n\\section{Conclusion}\nThe  coincident detection of the neutrino event, IceCube-170922A with the gamma ray flaring blazar, TXS 0506+056 provide support to the acceleration of cosmic rays in the blazar jet in diffusive shock acceleration process \\cite{IceCube18a}. In the framework of proton blazar model, our findings suggest that relative contributions to the total jet power of cold protons, accelerated protons, magnetic field, and accelerated electrons, obtained on the basis of charge neutrality, can explain both the low and high energy bump of the multi-wavelength EM SED and also the observed neutrino event, IceCube-170922A from the flaring blazar, TXS 0506+056 consistently. We find that maximum energy of cosmic ray particle achievable in the blazar is one order less then the ankle energy of cosmic ray energy spectrum or $2\\times 10^{17}$ eV in observer frame, is required to explain consistently the observed gamma ray and the neutrino signal from the source. The upcoming gamma-ray experiments like CTA \\cite{Ong17} and LHAASO \\cite{Liu17}, which are much sensitive up to 100 TeV energies, may provide clearer picture regarding the physical origin of gamma rays if more events like TXS 0506+056 are detected in future.\n\nThe gamma ray flux in the quiescent state of the source TXS 0506+056 is smaller by an order or so. Such a fact disfavor the cloud-jet interaction model as in the absence of cloud the gamma ray flux should decrease substantially. One may argue that the quiescent state gamma ray flux is due to inverse Compton process by relativistic electrons. But a fine tuning is needed to produce the exactly same kind of shape and same peak position of the second hump of EM SED in both enhanced and quiescent state if two different processes (hadronic and inverse Compton) are invoked to explain the observations. Recently, IceCube collaboration re-analyzed their historical data and reported significantly an evidence for a flare of 13 muon-neutrino events in the direction of TXS 0506+056 between September 2014 and March 2015 \\cite{IceCube18a}. Surprisingly, the blazar TXS 0506+056 was found to be in the quiescent state of both the radio and GeV emission at the arrival time window of such a neutrino flare \\cite{Padovani18}. Such an observation favors hadronic interaction mechanism for the production of observed high energy gamma rays and as well as neutrinos for both low and flaring state of the blazar. More elaborate studies are required to understand the production mechanism of the muon-neutrino events from TXS 0506+056 in the quiescent state.\n\n\\section*{Acknowledgments}\nThe authors would like to thank an anonymous reviewer for insightful comments and very useful suggestions that helped us to improve and correct the manuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}