diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpqsc" "b/data_all_eng_slimpj/shuffled/split2/finalzzpqsc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpqsc" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWe present combinatorial rules (one theorem and two conjectures)\nconcerning three bases of ${\\sf Pol}={\\mathbb Z}[x_1,x_2,\\ldots]$. \n\nConsider a basic question (studied for example in \\cite{Poly}):\n\\begin{quotation}\nHow does one\n lift properties of the ring $\\Lambda$ of symmetric functions (and its Schur basis)\nto the entirety of ${\\sf Pol}$?\n\\end{quotation}\nThe bases below lift the Schur polynomials. However, one wishes to analogize\nthe relationship in $\\Lambda$ between rules for Schur polynomials and \nLittlewood-Richardson rules. For these bases, no \nrule has yet provided a parallel, explaining a desire for alternative forms.\n\nFirst, we prove a ``splitting'' rule for the basis of\n\\emph{key polynomials} $\\{\\kappa_{\\alpha} | \\alpha\\in {\\mathbb Z}_{\\geq 0}^{\\infty}\\}$, thereby establishing a new positivity theorem about them. This family was introduced by \\cite{Demazure} and first studied combinatorially in \\cite{LS2,LS1}. Combinatorial rules for their monomial\nexpansion are known, see, e.g., \\cite{LS2, LS1, Reiner.Shimozono, HHL}. Our rule refines \n\\cite[Theorem 5(1)]{Reiner.Shimozono} and is compatible with the splitting rule \\cite[Corollary~3]{BKTY} for\nthe basis of \\emph{Schubert polynomials} $\\{{\\mathfrak S}_w |w\\in S_{\\infty}\\}$.\n\nSecond, we investigate a basis\n$\\{\\Omega_{\\alpha}|\\alpha\\in {\\mathbb Z}_{\\geq 0}^{\\infty}\\}$ \ndefined by \\cite{Lascoux:trans}\nthat deforms the key basis. By extending the \\emph{Kohnert moves} of \\cite{Kohnert} we conjecturally\ngive the first combinatorial rule for the $\\Omega$-polynomials.\n\nThird, in \\cite{Kohnert}, the Kohnert moves were used to conjecture the first combinatorial\nrule for Schubert polynomials (a proof was later presented in \\cite{Winkel}). Similarly, we use the extended Kohnert moves\nto give a conjecture for the basis of \\emph{Grothendieck polynomials} $\\{{\\mathfrak G}_w |w\\in S_{\\infty}\\}$ \\cite{LasSch2}. \nThis rule appears significantly different than earlier (proved) rules, such as those in \\cite{FK, Lascoux:trans, BKTY:II, LRS}. \n\n\\subsection{Splitting key polynomials}\n\n\nLet $S_{\\infty}$ be the group of permutations of ${\\mathbb N}$ with finitely many non-fixed points.\nThis acts on ${\\sf Pol}$ by permuting the variables.\nLet $s_i$ be the simple transposition interchanging $x_i$ and $x_{i+1}$. The {\\bf divided difference operator} acts on ${\\sf Pol}$ by\n\\[\\partial_i=\\frac{1-s_i}{x_i-x_{i+1}}.\\]\nDefine the {\\bf Demazure operator} by setting\n\\[\\pi_i(f)=\\partial_i(x_i \\cdot f ), \\mbox{ \\ for $f\\in {\\sf Pol}$.}\\]\n\nFor $\\alpha=(\\alpha_1,\\alpha_2,\\ldots)\\in {\\mathbb Z}_{\\geq 0}^\\infty$, the {\\bf key polynomial} $\\kappa_{\\alpha}$ is\n\\[\\kappa_{\\alpha}=x_1^{\\alpha_1}x_2^{\\alpha_2}\\cdots, \\ \\mbox{\\ if $\\alpha$ is weakly decreasing.}\\]\nOtherwise,\n\\[\\kappa_{\\alpha}=\\pi_i(\\kappa_{\\widehat \\alpha}) \\mbox{\\ where $\\widehat\\alpha=(\\ldots,\\alpha_{i+1},\\alpha_i,\\ldots)$ and\n$\\alpha_{i+1}>\\alpha_{i}$.}\\]\nSince the leading term of $\\kappa_{\\alpha}$ is $x_1^{\\alpha_1}x_2^{\\alpha_2}\\cdots$, the key polynomials\nform a ${\\mathbb Z}$-basis of ${\\sf Pol}$.\n\nThe key polynomials lift the Schur polynomials: when\n\\begin{equation}\n\\label{eqn:kappaequalsschurcond}\n\\alpha=(\\alpha_1,\\alpha_2,\\ldots,\\alpha_t,0,0,0,\\ldots), \\mbox{\\ where $\\alpha_1\\leq \\alpha_2\\leq\\ldots\\leq\\alpha_t$, then}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:kappaequalsschur}\n\\kappa_{\\alpha}=s_{(\\alpha_t,\\cdots,\\alpha_2,\\alpha_1)}(x_1,\\ldots,x_t).\n\\end{equation}\n\nA {\\bf descent} of $\\alpha$ is an index $i$ such that $\\alpha_{i}\\geq \n\\alpha_{i+1}$; a {\\bf strict descent} is an index\n$i$ such that $\\alpha_{i}>\\alpha_{i+1}$. Fix descents \n$d_1 0 , \\min T_2 > d_1, \\min T_3 > d_2, \\ldots, \\min T_k> d_{k-1}$; and\n\\item ${\\tt row}(T_1)\\cdot {\\tt row}(T_2)\\cdots {\\tt row}(T_k)$ is a reduced word of $w[\\alpha]$ such that\\\\\n${\\tt EGLS}({\\tt row}(T_1)\\cdot {\\tt row}(T_2)\\cdots {\\tt row}(T_k))=T[\\alpha]$.\n\\end{itemize}\n\\end{Theorem}\n\nWhen $d_j=j$ for all $j\\geq 1$, Theorem~\\ref{claim:main} specializes to an instance\nof the monomial expansion formula \\cite[Theorem~5(1)]{Reiner.Shimozono} for $\\kappa_{\\alpha}$ (restated as\nTheorem~\\ref{thm:RS} below). Also, when\n(\\ref{eqn:kappaequalsschurcond}) holds, $k=1$, $d_1=t$ and thus\nTheorem~\\ref{claim:main} gives (\\ref{eqn:kappaequalsschur}).\n\n\\begin{Example}\n\\label{exa:ispos}\nThe (strict) descents of $\\alpha=(1,3,0,2,2,1)$ are $d_1=2, d_2=5$, and\n\\begin{multline}\\nonumber\n\\kappa_{1,3,0,2,2,1}=s_{3,2}(x_1,x_2)s_{2,1,1}(x_3,x_4,x_5)+s_{3,2}(x_1,x_2)s_{2,1}(x_3,x_4,x_5)s_{1}(x_6)\\\\ \\nonumber\n+s_{3,1}(x_1,x_2)s_{2,2}(x_3,x_4,x_5)s_{1}(x_6)+s_{3,1}(x_1,x_2)s_{2,2,1}(x_3,x_4,x_5).\\nonumber\n\\end{multline}\nexhibits the claimed non-negativity of Theorem~\\ref{claim:main}. \n\nAlso, $w[\\alpha]=2516743$ (one line notation)\nand $T[\\alpha]=\\tableau{1&3&4\\\\2&5\\\\4&6\\\\5\\\\6}$.\nThus,\n${\\mathcal E}_{(3,2),(2,1,1),\\emptyset}^{(1,3,0,2,2,1)}=\n{\\mathcal E}_{(3,2),(2,1),(1)}^{(1,3,0,2,2,1)}=\n{\\mathcal E}_{(3,1),(2,2),(1)}^{(1,3,0,2,2,1)}=\n{\\mathcal E}_{(3,1),(2,2,1),\\emptyset}^{(1,3,0,2,2,1)}=1$ are respectively witnessed by\n\\[\\left(\\tableau{1&3&4\\\\2&5},\\tableau{4&6\\\\5\\\\6},\\emptyset\\right),\n\\ \\left(\\tableau{1&3&4\\\\2&5},\\tableau{4&6\\\\5},\\tableau{6}\\right), \\ \n\\left(\\tableau{1&3&4\\\\2},\\tableau{4&5\\\\5&6},\\tableau{6}\\right), \\mbox{ \\ and \\ }\n\\left(\\tableau{1&3&4\\\\2},\\tableau{4&5\\\\5&6\\\\6},\\emptyset\\right).\\]\nFor example, for the leftmost sequence, ${\\tt EGLS}(43152\\cdot 6456 \\cdot \\emptyset)=T[\\alpha]$ holds.\n\\qed\n\\end{Example}\n\n\\subsection{The $\\Omega$ polynomials}\nA.~Lascoux \\cite{Lascoux:trans} defines $\\Omega_{\\alpha}$\nfor $\\alpha=(\\alpha_1,\\alpha_2,\\ldots) \\in {\\mathbb Z}_{\\geq 0}^{\\infty}$ by replacing $\\pi_i$ in the definition\nof the key polynomials with the operator defined by\n\\[{\\widetilde \\pi}_i(f)=\\partial_i(x_i(1-x_{i+1})f).\\]\nThe initial condition is\n$\\Omega_{\\alpha}=x_1^{\\alpha_1} x_2^{\\alpha_2} x_{3}^{\\alpha_3}\\cdots \n(=\\kappa_{\\alpha})$, if $\\alpha$ is weakly decreasing.\n\n\nThe {\\bf skyline diagram}\nis ${\\tt Skyline}(\\alpha)=\\{(i,y):1\\leq y\\leq \\alpha_i\\}\\subset {\\mathbb N}^2$.\nGraphically, it is a collection of columns $\\alpha_i$ high.\nFor instance,\n\\[{\\tt Skyline}(1,3,0,2,2,1)=\\left(\\begin{matrix}\n\\!.\\! & \\!+\\! & \\!.\\! & \\!.\\! &\\!.\\! & \\!.\\! \\\\\n\\!.\\! & \\!+\\! & \\!.\\! & \\!+\\! &\\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!.\\! & \\!+\\! &\\!+\\! & \\!+\\!\n\\end{matrix}\\right)\\]\n\nBeginning with ${\\tt Skyline}(\\alpha)$, {\\bf Kohnert's rule} \\cite{Kohnert} \ngenerates\ndiagrams $D$ by sequentially moving any $+$\nat the top of its column to the rightmost open position in its row and to its left. (The result of such a move\nneed not be the skyline of any $\\gamma\\in {\\mathbb Z}_{\\geq 0}^\\infty$.)\nLet $x^D=\\prod_i x_i^{d_i}$ be the column weight where $d_i$ is the number of $+$'s in column $i$\nof $D$. If the same $D$ results from a different sequence of moves, it only counts once. Kohnert's theorem states $\\kappa_{\\alpha}=\\sum x^D$, where the\nsum is over all such $D$. Extending this, we introduce:\n\n\\noindent\n{\\bf The $K$-Kohnert rule:} Each $+$ either moves\n as in Kohnert's rule, or stays in place \\emph{and} moves. In the latter case, mark\n the original position with a ``$g$''. The $g$'s are unmovable, \nbut a given $+$\n treats $g$ the same as other $+$'s when deciding if it can move,\nand to where. Diagrams with\n the same occupied positions but different arrangements of $+$'s and $g$'s are counted separately.\n\n\\begin{Example}\nBelow, we give all $K$-Kohnert moves one step from $D$:\n\\[D=\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! & \\!+\\! &\\!.\\! \\\\\n\\!.\\! & \\!+\\! & \\!+\\! & \\!+\\! & \\!+\\!\n\\end{matrix}\\right)\\mapsto\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! & \\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\! & \\!+\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! & \\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!g\\! & \\!+\\! & \\!+\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!+\\! & \\!g\\! & \\!.\\! &\\!.\\! \\\\\n\\!.\\! & \\!+\\! & \\!+\\! & \\!+\\! & \\!+\\!\n\\end{matrix}\\right),\n\\]\n\\[ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!+\\! & \\!g\\! & \\!g\\! &\\!.\\! \\\\\n\\!.\\! & \\!+\\! & \\!+\\! & \\!+\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! & \\!+\\! &\\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!+\\! & \\!+\\! & \\!.\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! & \\!+\\! &\\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!+\\! & \\!+\\! & \\!g\\!\n\\end{matrix}\\right).\n\\]\n\\end{Example}\n\nLet\n \\[J_{\\alpha}^{(\\beta)}=\\sum \\beta^{(\\small \\#\\mbox{$g$'s appearing in $D$})}x^D.\\]\n\\begin{Conjecture}\n\\label{conj:OmegaequalsJ}\n$J_{\\alpha}^{(-1)}=\\Omega_{\\alpha}$.\n\\end{Conjecture}\nConjecture~\\ref{conj:OmegaequalsJ} has been checked by computer, for a wide \nrange of cases up to $\\alpha$ being of size $12$, leaving us convinced.\nClearly, $J_{\\alpha}^{(0)}=\\kappa_{\\alpha}$, by Kohnert's theorem.\n\n\\begin{Example} Let $\\alpha=(1,0,2)$. Then the diagrams contributing to $J_{(1,0,2)}$ are:\n\\[{\\tt Skyline}(1,0,2)=\\left(\\begin{matrix}\n\\!.\\! & \\!.\\! & \\!+\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!.\\! & \\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!.\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!.\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!.\\! & \\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!.\\!\n\\end{matrix}\\right);\n\\]\n\\[\\left(\\begin{matrix}\n\\!+\\! & \\!g\\! & \\!.\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!g\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!.\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!g\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!.\\! & \\!+\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!g\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!.\\! & \\!+\\! & \\!g\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right), \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!.\\! & \\!g\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right);\n\\left(\\begin{matrix}\n\\!+\\! & \\!g\\! & \\!.\\! \\\\\n\\!+\\! & \\!+\\! & \\!g\\!\n\\end{matrix}\\right); \\\n\\left(\\begin{matrix}\n\\!+\\! & \\!g\\! & \\!g\\! \\\\\n\\!+\\! & \\!.\\! & \\!+\\!\n\\end{matrix}\\right).\\]\nThus\n\\begin{multline}\\nonumber\nJ_{(1,0,2)}=(x_1 x_3^2 + x_1 x_2 x_3 + x_1^2 x_3 +x_1^2 x_2\n+x_1 x_2^2)\\\\ \\nonumber\n-(x_1^2 x_2 x_3 + x_1^2 x_2^2 +x_1^2 x_2 x_3\n+x_1 x_2^2 x_3 + x_1 x_2 x_3^2\n+x_1^2 x_3^2)+(x_1^2 x_2^2 x_3 + x_1^2 x_2 x_3^2).\n\\end{multline}\n\nThe lowest degree homogeneous\ncomponent of $\\Omega_{\\alpha}$ is $\\kappa_{\\alpha}$. Hence any $f\\in {\\sf Pol}$ is a\n possibly \\emph{infinite} linear combination of the $\\Omega_{\\alpha}$'s. Finiteness is\nasserted in \\cite[Chapter 5]{Poly}. We show in Section~4.2 that\nthe $J_{\\alpha}$'s also form a (finite) basis. \n\n\\subsection{Grothendieck polynomials}\nThe {\\bf Grothendieck polynomial} \\cite{LasSch2} is defined using the {\\bf isobaric divided difference\noperator} whose action on $f\\in {\\sf Pol}$ is given by:\n\\[\\pi_i(f)=\\partial_i((1-x_{i+1})f).\\]\nDeclare ${\\mathfrak G}_{w_0}(X)=x_1^{n-1}x_2^{n-2}\\cdots x_{n-1}$ \nwhere $w_0$ is the long element in $S_n$. Set ${\\mathfrak G}_w(X)=\\pi_i({\\mathfrak G}_{ws_i})$ \nif $i$ is an ascent of $w$.\nThe Grothendieck polynomials are known to lift $\\{s_{\\lambda}\\}$ to ${\\sf Pol}$.\n\nOne has \n${\\mathfrak G}_w={\\mathfrak S}_w+\\mbox{(higher degree terms)}$.\nWe now state the A.~Kohnert's conjecture \\cite{Kohnert} for ${\\mathfrak S}_w$.\nThe {\\bf Rothe diagram} is ${\\tt Rothe}(w)=\\{(x,y)|y2$ appears, we bump this $a_j+t$ to the next column to the right, replacing it with $a_j$.\nThe same holds if $a_j+1$ appears but not $a_j$. Finally, if both $a_j+1$ and $a_j$ already appear, we insert\n$a_j+1$ into the next column to the right. Since ${\\bf a}$ is assumed to be reduced, the above enumerates all possibilities. Finally at step $j$ a new box is created at a corner; in what will be $U$ we place $i_j$ .\n\nMildly abusing terminology, let ${\\tt EGLS}({\\bf a})=T$. \n\n\\excise{\\noindent\n{\\sf Description of $K_{-}^{0}(T)$ using Edelman-Greene:} For completeness we describe the computation of $K_{-}^0(T)$ in terms of Edelman-Greene insertion. Our description is different than, but equivalent to the definition from \\cite{Reiner.Shimozono}. The first column of $K_{-}^0(T)$ is the same as the first column of $T$. Suppose\nthe first $j$ columns of $K_{-}^0(T)$ have been determined. To determine the $(j+1)$-st column suppose the labels in column $j+1$ of $T$ are $x_1y_{m-1}>\\ldots>y_1$. Now continue by inserting these labels into column $j-1$ of $T$ etc.\n\nSince we will not actually need the details of this algorithm in our proof, we content ourselves with an example.\n\\begin{Example}\nIf $T=\\tableau{1&2&3\\\\2&3\\\\4}$ then $K_{-}^0(T)=\\tableau{1&1&1\\\\2&2\\\\4}$. To obtain the second column we reverse insert $3$ and $2$ into the first column of $T$\ngiving $2$ and $1$ respectively. To obtain the third column we reverse insert $3$ into the second column of $T$, outputting $2$. This $2$ is reverse inserted\ninto the first column, giving $1$.\\qed}\n\n\\end{Example}\n\\subsection{Formulas for Schubert polynomials}\nA stable compatible pair $({\\bf a},{\\bf i})$ is a {\\bf compatible pair for $w$}\nif in addition to (cs.1) and (cs.2) the following holds:\n\\begin{itemize}\n\\item[(cs.3)] $i_j\\leq a_j$.\n\\end{itemize}\nLet ${\\tt Compatible}(w)$ be the set of compatible sequences for $w$.\nA rule of \\cite{BJS} states:\n\\begin{equation}\n\\label{eqn:BJS}\n{\\mathfrak S}_w(X)=\\sum_{({\\bf a},{\\bf i})\\in {\\tt Compatible}(w)}{\\bf x}^{\\bf i}.\n\\end{equation}\n\nA {\\bf descent} of $w$ is an index $j$ such that $w(j)>w(j+1)$. Let ${\\tt Descents}(w)$ be the set of\ndescents of $w$. The following is \\cite[Corollary~3]{BKTY}:\n\n\\begin{Theorem}\n\\label{thm:BKTY}\nLet $w \\in S_n$ and suppose ${\\tt Descents}(w)\\subseteq \\{d_10, \\min T_2>d_1,\\ldots,\\min T_k>d_{k-1}$; and\n\\item[(iii)] ${\\tt row}(T_1)\\cdots {\\tt row}(T_k)$ is a reduced word of $w$.\n\\end{itemize}\n\\end{Theorem}\n\nAssume for the remainder of the proof that\n\\begin{equation}\n\\label{eqn:asspt}\n{\\tt Descents}(w)\\subseteq \\{d_11$.\n\n\\section{Comparing the ensembles}\n\nThe density of states of the GOE ensemble follows the famous\nsemicircle law, while the density of states of the TBRE is\nGaussian for large enough particle number, $n$, and orbital\nnumber, $m$, \\cite{brody81}; its width depends on $\\lambda$. In\norder to compare different ensembles, we restrict our statistical\nanalysis to a small energy interval with a constant level density\nat the center of the real spectrum of the complex eigenvalues of\n${\\cal H}$. This interval should be small enough in order to\nneglect the energy-dependent difference of the density of states\namong the ensembles, but large enough with respect to the widths\nin order to contain a statistically meaningful number of\nresonances.\n\nFor a model with a finite resonance number, it is important to avoid\nedge effects, see discussion in \\cite{moldauer67,moldauer75}. The\nenergy interval subject to statistical analysis should be also at a\ndistance of at least several widths away from the edges. A rough\nestimate \\cite{celardo1} goes as follows: for $M$ equivalent\nchannels, $\\Gamma\/D \\propto M$, and the distance from the center to\nthe edges is $ND\/2$, then $ND\/2\\gg\\Gamma\/D$ that implies $M\/N \\ll\n1\/2$. This shows that the ratio of the number of channels to that of\nresonances must be small in order for the results to be model\nindependent. With this choice, the model will be essentially\nequivalent to an infinite resonance model with a constant level\ndensity, apart from a narrow interval around the critical value. For\n$\\kappa=1$, with an infinite resonance number, the average widths\nshould logarithmically diverge, in agreement with the\nMoldauer-Simonius expression \\cite{simonius74}. In our finite model,\nthe results become model-dependent in a narrow interval near\n$\\kappa=1$. Our approach is still appropriate for a comparison with\nthe predictions of Ericson fluctuation theory derived for an\ninfinite resonance model with a constant level density.\n\nThe results of numerical simulations presented below refer to the\ncase of $N=924$ internal states and $M$ equiprobable channels,\n$\\kappa^{a}=\\kappa$. The maximum value of $M$ we considered is\n$M=25$ so that $M\/N \\approx 2\\cdot 10^{-2}$. For any value of\n$\\kappa$, we have used a large number of realizations of the\nHamiltonian matrices, with further averaging over energy.\n\n\\section{Ericson fluctuations}\n\nThe starting point of the conventional theory\n\\cite{ericson63,brink63,EMK66} can be summarized as follows. The\nscattering amplitude ${\\cal T}^{ab}(E)=\\langle{\\cal\nT}^{ab}(E)\\rangle + {\\cal T}^{ab}_{{\\rm fl}}(E)$ is divided into\ntwo parts, an average one, $\\langle{\\cal T}^{ab}(E)\\rangle$, and a\nfluctuating one, ${\\cal T}^{ab}_{{\\rm fl}}(E)$, with\n\\begin{equation}\n\\langle{\\cal T}^{ab}_{{\\rm fl}}(E)\\rangle =0. \\label{21}\n\\end{equation}\nNote that in our statistical model we have $\\langle {\\cal T}_{{\\rm\ninel}} \\rangle=0$ for inelastic channels, while\n\\begin{equation}\n\\langle {\\cal T}_{{\\rm el}} \\rangle=-i(1-\\langle S\n\\rangle)=-2i\\,\\frac{\\kappa} {1+\\kappa} \\label{39}\n\\end{equation}\nfor elastic channels.\n\nWith statistical independence of poles (resonance energies) and\nresidues (resonance amplitudes), $z_r\\equiv{\\cal A}_r^a \\tilde{{\\cal\nA}}_r^b$, we obtain $z_r=\\langle z_r \\rangle+ \\delta z_r$, so that\n\\begin{equation}\n{\\cal T}^{ab}(E)= \\sum_r \\frac{\\langle z_r \\rangle+\\delta\nz_r}{E-E_r+i \\Gamma_r\/2}. \\label{22}\n\\end{equation}\nIn the regime of overlapping resonances, $\\langle\\Gamma\\rangle > D$,\nand assuming all widths of the same order, $\\Gamma_r \\sim \\langle\n\\Gamma \\rangle$, the average part can be computed similarly to Eq.\n(\\ref{16}), substituting the sum by the integral,\n\\begin{equation}\n\\int\\frac{\\rho(E_r)\\langle z_r\\rangle dE_r}{E-E_r+i\\langle \\Gamma\n\\rangle \/2} \\approx -i \\frac{\\pi \\langle z_r \\rangle}{D},\n\\label{23}\n\\end{equation}\nwhere a constant level density, $\\rho(E_r)=1\/D$, is assumed.\n\nThe average cross section, $\\sigma= |{\\cal T}|^2$, also can be\ndivided into two contributions,\n\\begin{equation}\n\\langle\\sigma\\rangle=\\langle|{\\cal T}|^2\\rangle=|\\langle {\\cal T}\n\\rangle|^2+\\langle|{\\cal T}_{{\\rm fl}}|^2\\rangle. \\label{24}\n\\end{equation}\nThe two terms in Eq. (\\ref{24}) are interpreted as\n\\begin{equation}\n\\langle\\sigma\\rangle=\\langle\\sigma_{{\\rm dir}}\\rangle + \\langle\n\\sigma_{{\\rm fl}}\\rangle, \\label{25}\n\\end{equation}\nwhere the direct reaction cross section, $\\langle \\sigma_{{\\rm dir}}\n\\rangle$, is determined by the average scattering amplitude only,\nwhile $\\langle \\sigma_{{\\rm fl}} \\rangle$ is the fluctuational cross\nsection (also called in the literature the compound nucleus cross\nsection) that is determined by the fluctuational scattering matrix.\n\nFor overlapping resonances, $\\langle \\Gamma \\rangle > D$, the\nfollowing conclusions were derived concerning the scattering\namplitude and the statistical properties of the cross sections.\n\n(A). {\\sl The average fluctuational cross section} \\cite{ericson63}.\nAssuming that $\\Gamma_r \\approx \\langle\\Gamma\\rangle$ for a large\nnumber of channels, i.e. fluctuations of the widths around their\naverage value are small, ${\\rm Var}(\\Gamma)\/\\langle\\Gamma \\rangle^2\n\\ll 1$, the average fluctuational cross section,\n$\\langle\\sigma_{{\\rm fl}} \\rangle=\\langle{\\cal T}_{{\\rm fl}}{\\cal\nT}^*_{{\\rm fl}}\\rangle$, can be written as\n\\begin{equation}\n\\langle\\sigma_{{\\rm fl}}\\rangle=\\left\\langle\\sum_{rr'}\\frac{\\delta\nz_r^{\\ast} \\delta z_{r'}}{(E-E_r+i\/2\\langle\\Gamma\\rangle)(E-E_r'-i\/2\n\\langle \\Gamma \\rangle)}\\right \\rangle, \\label{26}\n\\end{equation}\nwhere the substitution $\\Gamma_r\\approx\\langle\\Gamma \\rangle$ was\nused. Now the averaging over energy is applied,\n\\begin{equation}\n\\langle F(E) \\rangle \\Rightarrow \\frac{1}{\\Delta E}\\, \\int dE\\,\nF(E). \\label{27}\n\\end{equation}\nThe integration leads to\n\\begin{equation}\n\\langle\\sigma_{{\\rm fl}}\\rangle= \\frac{2\\pi i}{\\Delta E}\\sum_{rr'}\n\\frac{\\delta z_r^{\\ast} \\delta z_{r'}}{E_{r'}-E_r+i\\langle\\Gamma\n\\rangle}.\n \\label{28}\n\\end{equation}\nNow we assume that $\\delta z_{r}$ are uncorrelated random quantities\nwith the statistics independent of $r$, $\\langle \\delta\nz_{r}^{\\ast}\\delta z_{r'}\\rangle=\\delta_{rr'}\\langle |\\delta\nz|^{2}\\rangle$. The absence of correlations between the amplitudes\n$\\delta z_r$ gives\n\\begin{equation}\n\\langle\\sigma_{{\\rm fl}} \\rangle = \\frac{2 \\pi}{D} \\frac{\\langle\n|\\delta z|^2 \\rangle}{\\langle \\Gamma \\rangle}, \\quad D=\\frac{\\Delta\nE}{N}. \\label{29}\n\\end{equation}\n\n(B). {\\sl Variance of the cross section}, ${\\rm Var}(\\sigma)=\\langle\n\\sigma^2 \\rangle- \\langle \\sigma \\rangle^2$. The derivation can be\nperformed under more general assumptions \\cite{ericson60} than those\nused for the analysis of average cross section. If the quantities\n$\\delta z_r \/(E- {\\cal E}_r)$ are independent complex variables,\nthen ${\\cal T}$ is Gaussian distributed, that is ${\\cal T}= \\xi +i\n\\eta $, where both $\\xi$ and $\\eta$ are Gaussian random variables\nwith zero mean. This is due to the fact that for $\\langle \\Gamma\n\\rangle \\gg D$ both $\\xi$ and $\\eta$ are the sums of a large number\nof random variables. In the conventional theory it is also assumed\nthat $\\xi$ and $\\eta$ have equal variance.\n\nThen for the fluctuating cross section we have\n\\begin{equation}\n\\langle\\sigma_{{\\rm fl}}^2 \\rangle= \\langle |{\\cal T}_{{\\rm fl}}|^4\n\\rangle= \\langle {\\cal T}_{{\\rm fl}}{\\cal T}_{{\\rm fl}}^*{\\cal\nT}_{{\\rm fl}}{\\cal T}_{{\\rm fl}}^*\\rangle= 2\\langle\\sigma_{{\\rm fl}}\n\\rangle^2. \\label{30}\n\\end{equation}\nIn a more general case when $\\langle {\\cal T} \\rangle \\ne 0$,\n\\begin{equation}\n{\\rm Var}(\\sigma)=\\langle\\sigma_{{\\rm fl}}\\rangle \\left( 2 \\langle\n\\sigma_{{\\rm dir}}\\rangle+\\langle\\sigma_{{\\rm fl}}\\rangle \\right).\n \\label{31}\n\\end{equation}\n\n(C). {\\sl The correlation function of the scattering amplitudes} is\ndefined as\n\\begin{equation}\nc(\\epsilon)=\\langle{\\cal T}(E+\\epsilon){\\cal T}^*(E)\\rangle -\n|\\langle{\\cal T}(E)\\rangle|^2=\\langle{\\cal T}_{{\\rm fl}}(E+\\epsilon)\n{\\cal T}_{{\\rm fl}}^*(E) \\rangle. \\label{32}\n\\end{equation}\nEvaluating $c(\\epsilon)$ under the same assumptions as for the\naverage cross sections, one obtains,\n\\begin{equation}\nc(\\epsilon)=\\langle\\sigma_{{\\rm fl}}\\rangle\\,\\frac{\\langle \\Gamma\n\\rangle}{\\epsilon+i \\langle \\Gamma \\rangle}. \\label{33}\n\\end{equation}\n\n(D). {\\sl The cross section correlation function} is defined as\n\\begin{equation}\nC(\\epsilon)=\\langle \\sigma(E)\\sigma(E+\\epsilon)\\rangle- \\langle\n\\sigma(E) \\rangle^2. \\label{34}\n\\end{equation}\nTaking into account the Gaussian form of distribution for $\\cal T$\nand Eq.(\\ref {33}), one obtains that:\n\n(a) the normalized autocorrelation function of cross sections\nsatisfies the relation,\n\\begin{equation}\n\\frac{C(\\epsilon)}{C(0)}=\\frac{|c(\\epsilon)|^2}{|c(0)|^2};\n \\label{35}\n\\end{equation}\n\n(b) the correlation function has a Lorentzian form,\n\\begin{equation}\n\\frac{C(\\epsilon)}{C(0)}=\\frac{l^2}{l^2+\\epsilon^2}, \\label{36}\n\\end{equation}\nwhere the correlation length, $l$, is equal to the average width,\n\\begin{equation}\nl=\\langle\\Gamma\\rangle. \\label{37}\n\\end{equation}\n\nIn the following we compare the predictions (B)-(D) of the\nconventional theory of Ericson fluctuations with our numerical\nresults, paying special attention to the dependence on the\nintrinsic interaction strength, $\\lambda$. As for the part (A),\nsince there are no predictions for the quantity $|\\delta z|^2$\ndetermining the average fluctuational cross section (\\ref{29}),\nthe comparison will be done with the Hauser-Feshbach, theory\nwidely used in the literature.\n\n\\section{Average cross section}\n\nIn this section we study how total and partial cross sections\ndepend on the continuum coupling, $\\kappa$, and intrinsic\ninteraction, $\\lambda$. For any value of $\\kappa$ we have used\n$N_r=30$ realizations of the Hamiltonian matrices. For each\nrealization we took into account only the interval [-0.2,0.2] of\nreal energy at the center of the spectrum.\n\nIt follows from Eq. (\\ref{13}) that the average total cross section\ndefined by the optical theorem,\n\\begin{equation}\n\\langle\\sigma_{{\\rm tot}}\\rangle=2 (1-{\\rm Re}\\,\\langle\nS\\rangle)=\\frac{4 \\kappa}{1+\\kappa}, \\label{40}\n\\end{equation}\ndepends only on the average scattering matrix and therefore is\nindependent of $\\lambda$ and $M$. Since $\\sigma_{{\\rm el}}=\n\\sigma_{{\\rm tot}}$ for $M=1$, the average elastic cross section\nis also independent of $\\lambda$ for the case of one channel. The\nsituation changes as we increase the number of channels.\n\nIn order to analyze the average elastic and inelastic cross section,\nwe single out the average scattering matrix elements in the standard\nform, $S^{ab}=\\langle S^{ab}\\rangle+S_{{\\rm fl}}^{ab}$, where\n$\\langle S^{ab}\\rangle=\\delta^{ab}\\langle S^{aa}\\rangle$ and\n$\\langle S_{{\\rm fl}}^{ab}\\rangle=0$. The average inelastic cross\nsection, $a\\neq b$, is given by\n\\begin{equation}\n\\langle \\sigma^{{\\rm ab}}\\rangle =\\langle|S^{ab}|^2\\rangle=\\langle\n|S_{{\\rm fl}}^{ab}|^2\\rangle. \\label{41}\n\\end{equation}\nThe average elastic cross section can be written as\n\\begin{equation}\n\\langle \\sigma^{aa} \\rangle = |1-\\langle\nS^{aa}\\rangle|^2+\\langle|S_{{\\rm fl}}^{aa}|^2\\rangle. \\label{42}\n\\end{equation}\nFollowing the literature we will call $\\langle|S_{{\\rm\nfl}}^{ab}|^2\\rangle$ the fluctuational cross section,\n$\\langle\\sigma_{{\\rm fl}}^{ab}\\rangle$. For $M$ equivalent\nchannels the fluctuational cross section can be expressed with the\nuse of the {\\sl elastic enhancement factor},\n\\begin{equation}\nF=\\frac{\\langle \\sigma_{{\\rm fl}}^{aa}\\rangle}{\\langle\n\\sigma_{{\\rm fl}}^{ab} \\rangle},\n\\label{43}\n\\end{equation}\nwhere $b\\neq a$. Indeed, in the case of equal channels, using the\nrelation,\n\\begin{equation}\n\\sigma_{{\\rm tot}}= \\sum_b \\sigma^{ab} = (M-1) \\sigma^{{\\rm\ninel}}+|1-S^{aa}|^2,\n \\label{44}\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\langle \\sigma_{{\\rm fl}}^{ab} \\rangle=\\frac{1-|\\langle\nS^{aa}\\rangle|^2}{F+M-1}=\\frac{T}{F+M-1}, \\label{45}\n\\end{equation}\nwhere $T$ is the transmission coefficient defined in Eq. (\\ref{20}),\nand\n\\begin{equation}\n\\langle \\sigma_{{\\rm fl}}^{aa} \\rangle=F\\langle \\sigma_{{\\rm\nfl}}^{ab} \\rangle=\\frac{FT}{F+M-1};\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, a \\neq b\n. \\label{46}\n\\end{equation}\n\nSince the transmission coefficient $T$ does not depend on\n$\\lambda$, the only dependence on $\\lambda$ in Eqs.~(\\ref{45}) and\n(\\ref{46}) is contained in the elastic enhancement factor $F$. The\nsame seems to be correct even when the channels are\nnon-equivalent, according to the results of Ref.~\\cite{muller87}.\nLeaving a detailed analysis of the elastic enhancement factor for\na separate study, here we point out that $F$ also depends on\n$\\kappa$. Specifically, with an increase of $\\kappa$ from zero,\nthe value of $F$ decreases, being confined by the interval between\n$3$ and $2$.\n\nFor the fluctuational inelastic cross section, with an increase of\nthe number of channels the dependence on the interaction strength\n$\\lambda$ disappears, see Fig.~\\ref{ERAVE} (upper panel). This is in\nagreement with the fact that in the limit of large $M$ we have $\n\\langle \\sigma_{{\\rm fl}}^{ab} \\rangle \\rightarrow T\/M$,\nindependently of $\\lambda$, see Eq.~(\\ref{45}). In contrast, the\nfluctuational elastic cross section manifests a clear dependence on\n$\\lambda$, see Fig.~\\ref{ERAVE} (lower panel). Thus, one can\ndirectly relate the value of the enhancement factor $F$ to the\nstrength $\\lambda$ of interaction between the particles. As one can\nsee, the more regular is the intrinsic motion, the higher is the\naverage cross section. Note that for a large number of channels the\n$\\lambda-$dependence of the elastic cross section is in agreement\nwith the estimate, $\\langle \\sigma_{{\\rm fl}}^{aa} \\rangle\n\\rightarrow FT\/M$, see Eq.~(\\ref{46}).\n\n\\begin{figure}[h!]\n\\includegraphics[width=8cm,angle=-90]{fig1.ps}\n\\vspace{-0.5cm} \\caption{(Color online) Fluctuational inelastic\n(upper panel) and fluctuational elastic (lower panel) cross sections\nas a function of $\\kappa$. Circles refer to the GOE case, pluses to\n$\\lambda=0$, crosses to $\\lambda=1\/30$, and squares to\n$\\lambda\\rightarrow\\infty$. The solid curves (top-down) correspond\nto the HF formula (\\ref{49}) with $F=2.5,\\,2.25,\\,2.0$ for\n$\\lambda=0,\\,1\/30$ and the GOE, respectively. The values $2.5$ and\n$2.25$ are found numerically using the definition (\\ref{43}).}\n\\label{ERAVE}\n\\end{figure}\n\nNow we compare our numerical results with the Hauser-Feshbach (HF)\nformula derived under quite general assumptions for both $a=b$ and\n$a \\neq b$ (see Ref. \\cite{brody81} and references therein),\n\\begin{equation}\n\\langle \\sigma_{{\\rm fl}}^{ab} \\rangle = (1+ \\delta_{ab})\\,\n\\frac{T^a T^b}{\\sum T^c}= (1+ \\delta_{ab})\\,\\frac{T}{M}.\n \\label{47}\n\\end{equation}\nHere the last expression corresponds to our case of equivalent\nchannels, $T^c=T$. As one can see, the HF formula predicts $F=2$,\nindependently of the interaction strength between the particles.\nThis formula was also derived in \\cite{agassi75} in the\noverlapping regime for the TBRE ensemble with the infinite\ninteraction, and in \\cite{lehmann95} for the GOE ensemble. For\nfinite number of channels in the limit $1\/M \\ll 1$, the corrected\nHF formula was derived in \\cite{agassi75}. For equivalent channels\nit reads,\n\\begin{equation}\n\\langle\\sigma_{{\\rm fl}}^{ab}\\rangle= (1+ \\delta_{ab})\\,\\frac{T}{M}\n\\left( 1-\\frac{1}{M}\\right). \\label{48}\n\\end{equation}\n\nOur data confirm that for the fluctuational {\\sl inelastic} cross\nsection the HF formula gives correct results for all values of\n$\\lambda$ in the case of large number of channels. The specific case\nof small number of channels, for which the HF is not valid, will be\ndiscussed elsewhere.\n\nOn the other hand, for the fluctuational {\\sl elastic} cross\nsection, our data show that the HF formula works only in the GOE\ncase and in the limit $\\lambda\\rightarrow\\infty$, see\nFig.~\\ref{ERAVE} (lower panel). At finite values of $\\lambda$ clear\ndeviations are seen. In order to describe the data, we modified the\nHF formula taking into account that the elastic enhancement factor\nvaries with $\\lambda$,\n\\begin{equation}\n\\langle \\sigma_{{\\rm fl}}^{ab} \\rangle=\\left[ 1+\\delta^{ab} (F-1)\n\\right] \\frac{T}{M} \\left( 1-\\frac{1}{M}\\right). \\label{49}\n\\end{equation}\nAs one can see, this expression gives a satisfactory description of\nthe data, with the numerically computed values of $F$. The problem\nof an analytical dependence of $F$ on the interaction strength\n$\\lambda$ remains open. To shed light on this problem, we performed\na specific study of the elastic cross section in dependence on\n$\\lambda$ for fixed value $\\kappa =0.8$ in the overlapping regime,\nsee Fig.~2. As one see, there is a sharp decrease of the cross\nsection in the transition from regular to chaotic intrinsic motion,\n$\\lambda \\approx \\lambda_{{\\rm cr}}$. This result is quite\ninstructive since it shows how the scattering properties are\ninfluenced by the onset of chaos in an internal dynamics. The\nnon-trivial point is that the analytical estimate of $\\lambda_{{\\rm\ncr}}$ was obtained for a closed system, $\\kappa =0$. However, even\nin the regime of a strong coupling to the continuum, $\\kappa = 0.8$,\nthis estimate gives a correct value for the interaction strength at\nwhich a drastic change of scattering properties occurs.\n\n\\vspace{-0.5cm}\n\\begin{figure}[h!]\n\\includegraphics[width=7cm,angle=-90]{fig2.ps}\n\\vspace{-0.5cm} \\caption{(Color online) Fluctuational elastic cross\nsection as a function of the interaction strength $\\lambda$ for\n$M=10$ and $\\kappa=0.8$ (connected circles). The horizontal line\nrefers to the GOE value, and dashed vertical line shows the critical\nvalue $\\lambda_{{\\rm cr}}$ for the transition to chaos in the TBRE,\nsee Eq. (\\ref{estimate}).} \\label{ELv0}\n\\end{figure}\n\n\\section{Fluctuations of widths and resonance amplitudes}\n\nHere we discuss the conventional assumption that for a large number\nof channels the deviations of the widths from their average are\nsmall, ${\\rm Var}(\\Gamma) \/\\langle\\Gamma\\rangle^2 \\ll 1$, and,\ntherefore, for analytical estimates one can set $\\Gamma_r\n\\approx\\langle\\Gamma\\rangle$, see Eq. (\\ref{26}). It is usually said\nin justification of this assumption \\cite{ericson63} that in the\noverlapping regime the width can be presented as a sum of partial\nwidths, $\\Gamma_r= \\sum_{c=1}^M \\Gamma_{r}^{c}$. Assuming that\nindividual partial widths obey the Potter-Thomas distribution, the\ntotal width is expected to have a $\\chi^2_M$ distribution, so that\n${\\rm Var} (\\Gamma)\/\\langle\\Gamma\\rangle^2 =2\/M$ is small for $M\\gg\n1$. In fact, it is sufficient to accept that the partial widths are\nindependent random variables; then ${\\rm Var}(\\Gamma) \\propto M$ and\n$\\langle\\Gamma\\rangle\\propto M$, so that ${\\rm Var}(\\Gamma)\/\\langle\n\\Gamma \\rangle^2 \\propto M^{-1}$.\n\nHowever, in our previous work \\cite{celardo1} we have showed that\nfor large values of $\\kappa$ the distribution of the widths\nstrongly differs from the $\\chi^2_M$ distribution. Our new data in\nFig.~3 give more details concerning this problem. These data were\nobtained for a large number $N_r=100$ realizations of the\nHamiltonian matrices, in order to have reliable results.\n\nThe data show that as $\\kappa$ increases the normalized variance,\n${\\rm Var}(\\Gamma)\/\\langle\\Gamma\\rangle^2$, also increases,\nremaining very large even for $M=20$. Moreover, the deviations\nfrom the expected $1\/M$ behavior are clearly seen signaling the\npresence of correlations in the partial widths. From Fig.~3 one\ncan also understand how the value of ${\\rm Var}(\\Gamma)\/\\langle\n\\Gamma\\rangle^2$ depends on the degree of intrinsic chaos\ndetermined by the parameter $\\lambda$. Specifically, for small\n$\\kappa$ there is no dependence on $\\lambda$ and ${\\rm\nVar}(\\Gamma)\/\\langle\\Gamma\\rangle^2$ decreases as $2\/M$ for all\nthe ensembles, as expected. However, as $\\kappa$ grows, the\ndependence on $\\lambda$ emerges: the weaker the intrinsic chaos\n(and, consequently, the more ordered is the intrinsic spectrum)\nthe larger are the width fluctuations.\n\n\\begin{figure}[h!]\n\\includegraphics[width=7.5cm,angle=-90]{fig3.ps}\n\\vspace{-0.5cm} \\caption{(Color online) Normalized variance of the\nwidth as a function of the number of channels $M$, for different\ncoupling strengths $\\kappa$ (connected symbols are the same as in\nFig.~\\ref{ERAVE}). While for small coupling, $\\kappa=0.01$, the\nvariance decreases with the number of channels very fast in\naccordance with the expected $\\chi^2$-distribution (dashed line),\nfor large couplings, $\\kappa=0.5$ and 0.9, the behavior is different\nfrom the $1\/M$-dependence.} \\label{VVG}\n\\end{figure}\n\nIn Ref. \\cite{SFT99} the widths distribution for the GOE ensemble\nwas found analytically for any number of channels in the limit of\n$N\\to\\infty$ and $M$ fixed. The general analytical result is given\nin terms of a complicated threefold integral. A simpler expression\nis obtained for a specific case $M=2$, for which the distribution\nof the widths is expressed as a double integral,\n\\begin{equation}\n\\langle y^2\\rangle=C\\int_{1}^{g}\\frac{d \\nu}{\\sqrt{\\nu^2-1}}\n\\int_{-1}^1 d \\mu \\frac{(1-\\mu^2)}{(\\nu+g-2\\mu)(\\nu-\\mu)^2},\n \\label{50}\n\\end{equation}\nwhere $C=1\/(2\\sqrt{g^2-1}) $, $y= \\pi \\Gamma\/D$, $g=(2\/T)-1$ and $T$\nis the transmission coefficient, so that from Eq. (\\ref{20}) we have\n$g=(1+\\kappa^2)\/2 \\kappa$, or $\\kappa= g \\pm \\sqrt{g^2-1}$. Note\nthat $\\nu+g-2\\mu>0$ and for $g =1$ we have $\\kappa=1$. The result of\nnumerical integration in Eq.~(\\ref{50}) is shown in Fig.~\\ref{M2int}\nby a solid curve. The comparison with the normalized variance of the\nwidths for the GOE case and $M=2$ (circles) shows a good agreement\nexcept for the vicinity of $\\kappa=1$. The difference at this point\nis due to the finite $N$ effects.\n\n\\vspace{-0.5cm}\n\\begin{figure}[h!]\n\\vspace{-0.5cm}\n\\includegraphics[width=7cm,angle=-90]{fig4.ps}\n\\vspace{-0.2cm} \\caption{(Color online) Numerical data for the\nnormalized variance of the widths vs $\\kappa$ for GOE and $M=2$\n(circles), in comparison with the result of numerical integration\nof Eq.~(\\ref{50}) (solid curve), and with Eq.~(\\ref{51}) (dashed\ncurve), see in the text. } \\label{M2int}\n\\end{figure}\n\nA specific interest is in the behavior of the variance at the\ntransition region, $\\kappa \\approx 1$. The analytical expression,\nderived from Eq. (\\ref{50}) for this region has the form,\n\\begin{equation}\n\\frac{{\\rm Var}(\\Gamma)}{\\langle\\Gamma\\rangle^2}=\\frac{2(2+\\pi)}\n{\\sqrt{2(g+1)} (g-1)} \\left( \\ln\\frac{g-1}{{g+1}} \\right)^2-1,\n \\label{51}\n\\end{equation}\nand is shown in Fig.~4 by the dashed curve. From the above\nrelation for $M=2$ one can obtain that the normalized variance\ndiverges as\n\\begin{equation}\n\\frac{{\\rm Var}(\\Gamma)}{\\langle\\Gamma\\rangle^2} \\propto\n\\frac{1}{(1-\\kappa)^2} \\left[\\ln(1-\\kappa)\\right]^2. \\label{52}\n\\end{equation}\n\nOur numerical simulations confirm that the divergence remains for\nany number of channels and for any value of $\\lambda$. Thus,\ncontrary to the traditional belief, the variance of widths does\nnot become small for a large number of channels. Another result is\nthat the assumption of the absence of correlations between the\nresonance amplitudes $\\delta z_r$ and the widths $\\Gamma_r$ seems\nto be incorrect in the region of a strong resonance overlap.\nIndeed, the data reported in Fig.~5 demonstrate that in contrast\nwith the case of weak coupling, $\\kappa = 0.001$, for a strong\ncoupling there are systematic correlations between $\\delta z_r$\nand $\\Gamma_r$. These correlations are increasing with an increase\nof the coupling strength, $\\kappa$, this effect is missed in the\nconventional description.\n\n\\vspace{-0.5cm}\n\\begin{figure}[h!]\n\\vspace{-0.5cm}\n\\includegraphics[width=7.5cm,angle=-90]{fig5.ps}\n\\caption{(Color online) Absolute squares of resonance amplitudes,\n$|\\delta z_r|^2$, versus the widths $\\Gamma_r$ for the GOE with\n$M=20$. As $\\kappa$ increases, the correlations between $|\\delta\nz_r|^2$ and $\\Gamma_r$ grow.}\n \\label{RA}\n\\end{figure}\n\n\\section{Statistics of cross sections}\n\n\\subsection{Distribution of fluctuational cross sections}\n\nAccording to the standard Ericson theory, the fluctuating\nscattering amplitude can be written as ${\\cal T}_{{\\rm\nfl}}^{ab}=\\eta+i\\xi$, where $\\eta$ and $\\xi$ are Gaussian random\nvariables with zero mean and equal variances. Since the\nfluctuating cross section is given by\n\\begin{equation}\n\\sigma_{{\\rm fl}}=|{\\cal T}_{{\\rm fl}}|^2=|\\eta|^2+|\\xi|^2,\n \\label{53}\n\\end{equation}\nthen $\\sigma_{{\\rm fl}}$ should have a $\\chi^2$ distribution with\ntwo degrees of freedom, that is an exponential distribution,\n\\begin{equation}\nP(x)= e^{-x}, \\quad x=\\frac{\\sigma_{{\\rm fl}}}{\\langle\\sigma_{{\\rm\nfl}}\\rangle}. \\label{54}\n\\end{equation}\nThis should be valid both for the elastic and inelastic cross\nsections.\n\n\\begin{figure}[h!]\n\\includegraphics[width=7.5cm,angle=-90]{fig6.ps}\n\\caption{(Color online) Distribution of the inelastic\nfluctuational cross section for the GOE and for the $\\lambda=0$\ncase, for $M=10;\\,25$ number of channels and fixed $\\kappa=0.9$. }\n\\label{ine}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[width=7.5cm,angle=-90]{fig7.ps}\n\\caption{(Color online) The same as Fig.6, but for the elastic\nfluctuational cross section. A clear difference from the exponential\ndistribution is seen in the tails.} \\label{el}\n\\end{figure}\n\nIn Figs. \\ref{ine} and \\ref{el} we show the distribution of\nfluctuational cross sections for the elastic and inelastic cross\nsections, with two different numbers of channels, $M=10$ and $M=25$.\nAnalyzing these data, one can draw the following conclusions. First,\nfor the inelastic cross section the data seem to follow the\npredicted exponential distribution. It should be noted, however,\nthat a more detailed analysis with the help of the $\\chi^2$-test\nreveals the presence of strong deviations.\n\nThe situation with the fluctuational elastic cross section is\ndifferent due to strong deviations from the exponential distribution\noccurring even for a quite large $M=25$. The fact that large\ndeviations from the conventional theory (for finite values of $M$)\nshould be expected in the elastic case were recognized also in\nRefs.~\\cite{DB1,DB2}. The comparison between $M=10$ and $M=25$ cases\nindicate that it is natural to assume that with a further increase\nof $M$ both the distributions will converge to the exponential one.\nIt is important to note that there is a weak dependence on the\ninteraction strength $\\lambda$ between the particles. This is\nconfirmed by a closer inspection of the data of Fig.~\\ref{el}.\nSpecifically, the data clearly show that there is a systematic\ndifference for the two limiting cases of zero and infinitely large\nvalues of $\\lambda$. Our results for the normalized variance (see\nbelow), indeed, confirm a presence of this weak dependence on\n$\\lambda$.\n\nThe above data for the distribution of the normalized fluctuational\ncross section may be treated as a kind of confirmation of the\nEricson fluctuation theory. However, it should be stressed that if\nwe are interested in the fluctuations of non-normalized cross\nsections (at least, for elastic cross sections), one should take\ninto account the dependence on $\\lambda$. Currently no theory allows\none to obtain the corresponding analytical results, even for the\nsituation where the number of channels is sufficiently large.\n\n\\subsection{Fluctuations}\n\nHere we compare our results for the variance of cross sections with\nthe Ericson fluctuations theory \\cite{ericson63}, and with more\nrecent results for the GOE \\cite{DB1,DB2}. According to the standard\npredictions, the variance of fluctuations of both elastic and\ninelastic cross sections,\n\\begin{equation}\n{\\rm Var}(\\sigma^{ab})=\\langle(\\sigma^{ab}-\n\\langle\\sigma^{ab}\\rangle)^2\\rangle, \\label{55}\n\\end{equation}\nis directly connected to the average cross sections by\nEq.~(\\ref{31}). It is useful to express the variance of the cross\nsections in terms of the scattering matrix. In our statistical model\nfor $a \\ne b$, $\\langle{\\cal T}^{ab}\\rangle=i\\langle\nS^{ab}\\rangle=0$. Therefore, the variance of the inelastic cross\nsection reads\n\\begin{equation}\n{\\rm Var}(\\sigma^{ab})=\\langle\\sigma_{{\\rm fl}}^{ab}\\rangle^2\n=\\langle|S^{ab}_{{\\rm fl}}|^2\\rangle^2. \\label{56}\n\\end{equation}\n\nFor the elastic scattering one can write $\\langle {\\cal T}^{aa}\n\\rangle=-i (1-\\langle S^{aa}\\rangle)$, so that $\\sigma_{{\\rm\ndir}}=|1-\\langle S^{aa} \\rangle|^2$, and $\\sigma_{{\\rm fl}}=\\langle\n|S^{aa}_{{\\rm fl}}|^2 \\rangle$. Therefore, for the variance of the\nelastic cross sections, one obtains,\n$$\n{\\rm Var}(\\sigma^{aa})=2 \\langle\\sigma_{{\\rm fl}}^{aa}\\rangle\n\\langle \\sigma_{{\\rm dir}}^{aa}\\rangle+\\langle\\sigma_{{\\rm\nfl}}^{aa}\\rangle^2 =\n$$\n\\begin{equation}\n=\\langle|S^{aa}_{{\\rm fl}}|^2\\rangle^2+2|1-\\langle\nS^{aa}\\rangle|^2 \\langle |S^{aa}_{{\\rm fl}}|^2 \\rangle, \\label{57}\n\\end{equation}\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=7.0cm,angle=-90]{fig8.ps}\n\\caption{(Color online) Variance of inelastic (upper panel) and\nelastic (lower panel) cross sections, see Eq.~(\\ref{25}), for $M=10$\nas a function of $\\kappa$ for different interaction strengths: the\nGOE (circles), $\\lambda=0$ (pluses), $\\lambda\\rightarrow\\infty$\n(squares), and $\\lambda=1\/30$ (crosses). For the comparison the\ntheoretical curve for $MT \\gg 1$, obtained for the GOE from\nEqs.~(\\ref{60},\\ref{61}) (solid curve) is also added. Note that the\nmaximum of ${\\rm Var}(\\sigma^{el})$ is shifted from $\\kappa=1$ due\nto the presence of direct processes.} \\label{fluctELINE}\n\\end{figure}\n\nThe discussed above conventional predictions and our analysis of the\naverage cross section in Sec. VI imply that the variances of cross\nsections depend on the intrinsic interaction strength $\\lambda$\nthrough the average cross sections. Our data for a relatively large\nnumber $M=10$ of channels, see Fig.~\\ref{fluctELINE}, indeed,\ncorrespond to this expectation. Since for the inelastic scattering\nthe average cross section does not depend on the interaction\nstrength, it is quite expected that the same occurs for the variance\nof the inelastic cross section. Our data confirm this expectation.\nOn the other hand, the variance of the elastic cross section reveals\na clear dependence on the value of $\\lambda$. As one can see, this\ndependence is quite strong in the region of strongly overlapped\nresonances, for $\\kappa \\approx 1$.\n\nLet us now compare our data with the exact expressions for the\nvariance of cross sections. To do this, it is convenient to express\nthis variance in terms of the scattering matrix,\n$$\n{\\rm Var}(\\sigma^{ab})=\\langle|S^{ab}_{{\\rm fl}}|^4\\rangle-\n\\langle|S^{ab}_{{\\rm fl}}|^2\\rangle^2+\n$$\n\\begin{equation}\n-\\delta^{ab}\\Bigl(2\\Bigl[(1-\\langle S^{aa\\ast}\\rangle)\n\\langle|S^{aa}_{{\\rm fl}}|^2 S^{aa}_{{\\rm fl}}\\rangle + {\\rm c.c.}\n\\Bigr]-2 |1-\\langle S^{aa}\\rangle|^2 \\langle|S^{ab}_{{\\rm fl}}|^2\n\\rangle\\Bigr). \\label{58}\n\\end{equation}\nComparing Eq. (\\ref{58}) with the standard predictions, Eqs.\n(\\ref{56},\\ref{57}), one can see that they are correct if:\n$$\n({\\rm i})\\;\\langle|S^{ab}_{{\\rm fl}}|^4\\rangle\n-2\\langle|S^{ab}_{{\\rm fl}}|^2\\rangle^2=0,\n$$\n\\begin{equation}\n({\\rm ii})\n\\;\\langle S_{{\\rm fl}}^{aa}|S^{ab}_{{\\rm fl}}|^2\\rangle=0.\n \\label{59}\n\\end{equation}\nThese properties are consistent with the Gaussian character of the\ndistribution for the fluctuational scattering matrix.\n\nThe analytical expressions for the variance of elastic and inelastic\ncross sections were obtained in Refs. \\cite{DB1,DB2} for the GOE\ncase, any number of channels and any coupling strength with the\ncontinuum. However, simple expressions were derived only for $MT \\gg\n1$. Even under such a condition, the analytical results show\ndeviations from the conventional assumptions. Specifically, it was\nfound,\n$$\n({\\rm i})\\; \\langle|S^{ab}_{{\\rm fl}}|^4\\rangle-2\\langle|S^{ab}_{{\\rm\nfl}}|^2\\rangle ^2=\n$$\n\\begin{equation}\n=(1+7\\delta^{ab})[6-4(T^a+T^b)+r_2] \\,\\frac{2(T^aT^b)^2}{(S_1+1)^3},\n \\label{60}\n\\end{equation}\nand\n\\begin{equation}\n({\\rm ii})\\; \\langle|S^{aa}_{{\\rm fl}}|^2 S^{aa}_{{\\rm fl}}\\rangle= -8\\langle\nS^{aa\\ast}\\rangle\\,\\frac{(T^a)^3}{(S_{1}+1)^2}, \\label{61}\n\\end{equation}\nwhere $S_1=\\sum T^c$, $S_2=\\sum (T^c)^2$ and $r_2=(S_2+1)\/(S_1+1)$.\n\nThe theoretical values for the variance of the cross sections\nobtained from Eqs.(\\ref{60},\\ref{61}) and from the HF formula,\nthrough Eq.(\\ref{58}), are shown in Fig.~\\ref{fluctELINE} by solid\ncurve. The agreement is good for the GOE case in the strong coupling\nregime, as expected. As we can see from Eqs. (\\ref{60},\\ref{61}),\nassumptions of Eq. (\\ref{59}) are valid for large $S_1$. In\nparticular it was shown in \\cite{DB1,DB2} that the ratio ${\\rm\nVar}(\\sigma_{{\\rm fl}})\/\\langle\\sigma_{{\\rm fl}}\\rangle^2$, being\nequal to one in standard theory, significantly differs from unity in\nthe range $10$, and correlation length, $l_\\sigma$, normalized to\nthe mean level spacing at the center of the spectra, versus $\\kappa$\nfor $M=1$ (upper panel) and $M=10$ (lower panel). Solid curves show\nthe MS-expression (\\ref{61}). Open circles refer to the GOE case,\npluses to $\\lambda=0$, squares to $\\lambda \\rightarrow\\infty$, and\ncrosses to $\\lambda=1\/30$, all for the normalized average width.\nFull triangles stand for the normalized correlation length at\n$\\lambda=1\/30$. The dashed line shows the Weisskopf relation\n(\\ref{60}).} \\label{924GD10}\n\\end{figure}\n\n\\vspace{-0.5cm}\n\\begin{figure}[h!]\n\\includegraphics[width=8cm,angle=-90]{fig11.ps}\n\\vspace{-1.0cm} \\caption{(Color online) Elastic-elastic\ncorrelation length of the cross section at different values of\n$\\lambda$ as a function of $\\kappa$ for $M=1$ and $M=10$; the GOE\ncase (connected circles), $\\lambda=0$ (pluses connected by a\nline), $\\lambda\\rightarrow\\infty$ (squares), and $\\lambda=1\/30$\n(crosses). The Weisskopf relation (\\ref{62}) is shown by dashed\ncurve. } \\label{Ccorr}\n\\end{figure}\n\nIn Fig. \\ref{Ccorr}, it is shown that, for a large number of\nchannels, the elastic correlation length is in agreement with\nEq.(\\ref{62}) for all values of the interaction strength, $\\lambda$.\nThe same occurs for the inelastic correlation length.\n\nThe Weisskopf relation (\\ref{62}) has been also derived in Ref.\n\\cite{agassi75} for small values of the ratio $m=M\/N$, in the\noverlapping regime for the TBRE ensemble with the infinite\ninteraction, as well as in Ref. \\cite{verbaarschot85} for the GOE\nensemble. In \\cite{lehmann95} the correlation function for the GOE\nensemble was computed also when $m=M\/N$ is not small, and the\ndeviations from Eq. (\\ref{62}) and from the Lorentzian form of the\ncorrelation function were found. This is not in contrast with our\nresults for the case of small ratio of $m$, see discussion in Sec.\nIV.\n\nThe fact that the correlation length is not equal to the average\nwidth was recognized long ago, see Refs. \\cite{moldauer75} and\n\\cite{brody81}. However, the statements based on the equality\n(\\ref{37}) still appear in the literature, see, for example, Ref.\n\\cite{abfalterer00}. The relation between the average resonance\nwidth and the transmission coefficient is given by the\nMoldauer-Simonius (MS) formula \\cite{moldauer67,simonius74},\n\\begin{equation}\nM \\ln(1-T)=-2 \\pi\\,\\frac{\\langle\\Gamma\\rangle}{D}. \\label{63}\n\\end{equation}\nIt can be seen from this expression and Eq. (\\ref{62}) that the\nequality $l=\\langle\\Gamma\\rangle$ is true only for small $T$.\n\n\\section{Conclusions}\n\nIn conclusion, we have studied the statistics of cross sections\nfor a fermion system coupled to open decay channels. For the first\ntime we carefully followed various signatures of the crossover\nfrom isolated to overlapping resonances in dependence on the\nstrength of inter-particle interaction modelled here by the\ntwo-body random ensemble. The study was performed for the simplest\nGaussian ensemble of decay amplitudes. Even in this limiting case,\nwhen these amplitudes were considered as uncorrelated with the\nintrinsic dynamics, we found significant dependence of reaction\nobservables on the strength of intrinsic interaction. We expect\nthis dependence to be amplified with realistic interplay of decay\namplitudes and internal wave functions. Such studies should be\nperformed in the future.\n\nA detailed comparison has been carried out of our results with\nstandard predictions of statistical reaction theory. The average\ncross section was compared with the Hauser-Feshbach formula for a\nlarge number of channels. In the inelastic case this description\nworks quite well in the overlapping resonance regime for any\ninteraction strength, while in the elastic case strong deviations\nhave been found if the intrinsic motion is not fully chaotic.\n\nThe study of Ericson fluctuation theory shows that the assumption\nthat the fluctuations of the resonance widths become negligible\nfor a large number of channels is wrong in the overlapping regime.\nWe found that the fluctuations of resonance widths increase with\nthe coupling to the continuum, and we gave evidence that the\nrelative fluctuation of the width (the ratio of the variance to\nthe square of the average width) diverges at $\\kappa=1$ for any\nnumber of channels. This should imply that for any number of\nchannels the differences from the standard theory should increase\nas $\\kappa$ increases.\n\nIn order to study the relationship between the variance of the\ncross section and its average value, it is necessary to take into\naccount the dependence of the average cross section on the\nintrinsic interaction strength $\\lambda$. Even when this is done,\nthe standard prediction about the variance of the cross section\nwas found to be a good approximation only for a very large number\nof channels. For $M$ between $10$ and $20$, where the Ericson\nprediction could be expected to be valid, consistent deviations\nhave been demonstrated.\nIn particular, the distribution of cross sections shows that the\nprobability of a large value of the cross section, mainly for the\nelastic case (or in the presence of direct reactions), can be well\nbelow conventional predictions.\n\nFinally, we have shown that, in agreement with previous studies, the\ncorrelation length differs from the average width for any number of\nchannels. On the other hand, the Weisskopf relation (\\ref{62}) that\nconnects the correlation length of the cross section to the\ntransmission coefficient, works, for a large number of channels, at\nany value of the intrinsic interaction strength $\\lambda$. In many\nsituations we have seen that increase of $\\lambda$ in fact\nsuppresses the fluctuations in the continuum. This can be understood\nqualitatively as a manifestation of many-body chaos that makes all\ninternal states uniformly mixed.\n\nOur results can be applied to any many-fermion system coupled to the\ncontinuum of open decay channels. The natural applications first of\nall should cover neutron resonances in nuclei, where rich\nstatistical material was accumulated but the transitional region\nfrom isolated to overlapped resonances was not studied in detail.\nThe interesting applications of a similar approach to molecular\nelectronics and electron tunneling spectroscopy can be found in the\nrecent literature \\cite{cacelli07,walczak07}. Other open mesoscopic\nsystems, for example, quantum dots and quantum wires, should be\nanalyzed as well in the crossover region. One can expect very\npromising designated studies for checking the statistical properties\nof resonances in such controllable experiments as those in microwave\ncavities and in acoustical chaos. Open boson systems in atomic traps\nalso can be an interesting object of future theoretical and\nexperimental studies.\n\n\\section{Acknowledgment}\n\nWe acknowledge useful discussion with D. Savin, T.~Kawano,\nV.~Sokolov and T. Gorin. The work was supported by the NSF grants\nPHY-0244453 and PHY-0555366. The work by G.P.B. was carried out\nunder the auspices of the National Nuclear Security Administration\nof the U.S. Department of Energy at Los Alamos National Laboratory\nunder Contract No. DE-AC52-06NA25396. F.M.I. acknowledges partial\nsupport by the CONACYT (M\\'exico) grant No~43730.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:Intro}Introduction}\nFirst-order phase transitions occur in a variety of systems and are characterized by a discontinuous first derivative in the free energy and phase coexistence at the transition temperature. These characteristics pose difficulties for typical canonical ensemble Monte Carlo simulations because the transition rate between the coexisting phases decreases exponentially in system size at the transition temperature leading to exponential slowing. \n\nExponential slowing can be substantially reduced using multi-canonical \\cite{BeNe92} or Wang-Landau \\cite{WaLa01,NoItWa11} methods. A related means for reducing exponential slowing is to simulate the system in the microcanonical ensemble~\\cite{MM07} rather than the canonical ensemble. For a temperature-driven, first-order phase transition, the coexisting phases will have distinct energies. \nConfigurations with energies between the coexisting values of the energy possess domains of both phases separated by an interface and are very improbable in the canonical ensemble. In the microcanonical ensemble, however, a droplet of a minority phase is stable at energies between the energies of the two coexisting phases, and so simulations can be carried out relatively efficiently at any energy in the coexistence region and exponential slowing is much reduced. Nonetheless, in the microcanonical ensemble, there remains two or more barriers associated with condensation\/evaporation \\cite{BiChKo02,NoItWa11,ScZiJa16} transitions at the ends of the co-existence regions and, depending on the geometry of the system, also transitions in the topology of the growing droplet~\\cite{LeZi90} so that exponential slowing cannot be entirely eliminated.\n\nIn this paper we introduce a class of microcanonical Monte Carlo annealing algorithms and show that they are effective for simulating the strong first-order transition in large-$q$ Potts models. These algorithms are related to the well-known simulated annealing~\\cite{KiGeVe83} algorithm but with two important differences. First, annealing is carried out in energy rather than temperature, that is, we do microcanonical annealing. Second, we introduce an additional resampling step in the annealing process in order to properly sample the equilibrium ensemble at every energy. \nOne of the algorithms in this class that is studied here is a microcanonical version of population annealing~\\cite{HuIb03,Mac10a,CaMa17}.\n\nThe purpose of annealing in the current setting differs from the usual objective of finding ground states or sampling thermal states for systems with rough free energy landscapes. Here, microcanonical annealing is used to collect data at all energies in order to use reweighting to efficiently extract results at the transition temperature in the canonical ensemble.\n\nThe microcanonical ensemble is difficult to simulate using Markov chain Monte Carlo (MCMC) because of the constraint that proposed moves are accepted only if they do not change the energy. Here we circumvent that problem by simulating the ensemble of all equally weighted configurations with energies equal to {\\em or below} a given energy ceiling, which we call the energy ceiling ensemble. It is straightforward to transform results from the energy ceiling ensemble to the microcanonical ensemble. An alternative approach to microcanonical Monte Carlo simulations is to introduce a kinetic energy term in the Hamiltonian and require that the total energy be fixed while the Monte Carlo moves change only the potential energy~\\cite{MM07, ScZiJa16}. This ``real'' microcanonical ensemble achieves the same objective of allowing a continuous range of potential energies for a fixed value of the energy control parameter. \n\nSimulated annealing \\cite{KiGeVe83} consists of initializing a system at high temperature where equilibration using MCMC is easy, and gradually cooling the system following an annealing schedule that specifies a sequence of temperatures and number of MCMC sweeps at each temperature. The goal of simulated annealing is to find the ground state and, in its standard form, it is not a suitable algorithm for sampling equilibrium states. \n{\\it Equilibrium} simulated annealing used here differs from standard simulated annealing in the choice of the starting configuration at the beginning of each annealing step. In standard simulated annealing, the initial configuration at a new temperature is the final configuration generated by the MCMC at the previous temperature. Note that this procedure is not well-defined in the energy ceiling ensemble since the last configuration generated by the MCMC may not be allowed at the new energy. In equilibrium simulated annealing, the new configuration is randomly sampled with appropriate weights for the new annealing step from the set of configurations generated by the MCMC at the previous annealing step. In the energy ceiling ensemble these weights are equal and the new state is randomly and uniformly chosen from the configurations generated at the previous energy that satisfy the new energy ceiling. Thus, to go from one annealing step to the next we ``resample'' (or, more correctly, ``subsample'') from the configurations generated at the previous step.\n\nIf multiple replicas of the system are simultaneously annealed, resampling can also be done among the set of replicas. The resulting scheme is population annealing \\cite{HuIb03,Mac10a,WaMaKa15,WaMaKa15b,AmMa18,BaMaWeHe18} and is an example of a sequential Monte Carlo \\cite{DoFrGo01} algorithm. Population annealing in the canonical ensemble is an effective tool for simulating equilibrium systems with rough free energy landscapes, such as spin glasses~\\cite{WaMaKa15b,WaMaMuKa17,AmMa18}. Canonical population annealing has been applied to the first-order transition in Potts models~\\cite{BaWeShJa17}. \nIn microcanonical population annealing, resampling consists of discarding replicas that do not satisfy the new energy ceiling and making copies of those that do satisfy the new ceiling so that the population size remains constant. `Microcanonical' annealing in density rather than energy has been used to study hard sphere fluids~\\cite{CaMa17}. Microcanonical population annealing is very similar to nested sampling~\\cite{MaStWaFr14}. Finally, we introduce a class of algorithms that interpolate between equilibrium simulated annealing and population annealing. In these ``hybrid\" annealing algorithms, multiple replicas are annealed and resampling is done from the union of time series of each replica. \n\nAnnealing in the energy ceiling ensemble provides a direct measurement of the entropy, which can be used to calculate the canonical free energy and partition function, thus giving access to canonical ensemble observables at any temperature. The entropy or free energy can also be used to perform weighted averages over several simulation runs, reducing both statistical and systematic errors of observables in the microcanonical or canonical ensembles~\\cite{Mac10a}. \n\nThe primary goal of the paper is to introduce the equilibrium microcanonical annealing framework and test the performance of these algorithms for strong first-order transitions. A secondary goal is to to obtain high precision results for the finite-size, two-dimensional, 20-state Potts model.\nThe paper is organized as follows. In Sec.\\ \\ref{sec:Algorithm} we present the microcanonical annealing algorithms used in the simulations. In Sec.\\ \\ref{sec:model_and_observables} we define the $q$-state Potts model and the observables studied in the simulations. Section \\ref{sec:simdet} gives details of the simulations and in Sec.\\ \\ref{sec:Results} we present results from the simulations. The paper closes with a discussion in Sec.\\ \\ref{sec:discussion}.\n\n\\section{\\label{sec:Algorithm} Equilibrium Microcanonical Annealing}\n\nIn this section we present a general framework for equilibrium microcanonical annealing. Two limiting cases of this framework are population annealing and equilibrium simulated annealing. In all cases we have an ``annealing schedule'' that consists of a sequence of energy ceilings $\\{\\ceiling^{(k)}\\}$ and number of MCMC sweeps, $n_s(E^{(k)})$ that are carried out at each energy ceiling. We study these algorithms in the context of the Potts model, which has a bounded, discrete set of energy levels. For such systems the annealing schedule is naturally chosen as the set of energy levels of the system and annealing is carried out from the highest energy level to the ground state. In the following discussion the parenthetical superscript `$(k)$' is suppressed.\n\nMicrocanonical annealing directly simulates the energy ceiling ensemble in which all configurations with energies less than or equal to the energy ceiling, $\\ceiling$ are equally probable. In general, microcanonical annealing works with a population of replicas of the system that represent the energy ceiling ensemble. Each annealing step is carried out in two stages. In the first stage a population of $R$ replicas of the system is updated using a MCMC updating procedure designed to equilibrate to the energy ceiling ensemble. During the course of this updating a pool of $\\kR$ replicas of the system is saved where, generally, $\\kR \\geq R \\geq 1$. \n\nThe MCMC procedure used here is a single-spin-flip algorithm. The proposed update consists of flipping a randomly chosen spin. Suppose that the initial spin state is $\\alpha$ and the final spin state is $\\gamma$ with energy $E_\\gamma$. The acceptance probability, $P(\\alpha\\rightarrow \\gamma)$ of the proposed move is given by,\n\\begin{equation}\n P(\\alpha\\rightarrow \\gamma)=\n \\begin{cases} \n 1 & E_{\\gamma} \\leq \\ceiling \\\\\n 0 & E_{\\gamma} > \\ceiling . \\\\\n \\end{cases}\n \\label{eq:update}\n\\end{equation}\nThis MCMC algorithm is carried out for $n_s(\\ceiling)$ sweeps on each replica, where a sweep consists of $N$ updates and $N$ is the number of spins in the system. Clearly, this algorithm satisfies detailed balance with respect to the energy ceiling ensemble. If, in addition, the algorithm is ergodic, that is, all configurations of the system can be reached from one another via a series of updates, then the algorithm will converge to the desired energy ceiling ensemble. However, it is evident that this algorithm is {\\it not} ergodic for all ceiling energies. For example, for Ising-Potts models with $\\ce$ sufficiently near the ground state, it is not possible to go from one ordered state to another and remain below the ceiling via single spin flip moves. Nonetheless, full coverage of configuration space and accurate equilibrium results can be obtained either by using a population of replicas of the system as is done in population annealing and\/or by combining multiple independent runs as is done in equilibrium simulated annealing. \n\nThe second stage of microcanonical annealing consists of resampling with replacement from the pool of $\\kR$ replicas with energy ceiling $\\ceiling$ in order to generate a population of $R$ replicas that is approximately in equilibrium at the next lower value of the energy ceiling, $\\ceiling^\\prime$. Note that if $\\kR$ is not sufficiently large the pool may contain no configurations with energy less than or equal to $\\ceiling^\\prime$, in which case the algorithm fails. \n\nThe entire microcanonical annealing procedure is as follows:\n\n\\begin{enumerate}\n\\item A population of $R$ independent replicas of the system is initialized in equilibrium with the energy ceiling $\\ceiling$, set to the highest energy level of the system. \n\\item $n_s(\\ce)$ sweeps of the MCMC procedure are performed on each member of the population and $\\kR$ configurations generated from these MCMC sweeps are saved in the configuration pool. \n\\item The energy ceiling is lowered to the next value $\\ce^\\prime$ in the annealing schedule ($\\ceiling^\\prime< \\ceiling$) and $R$ configurations with energy less than or equal to $\\ceiling^\\prime$ are randomly drawn with replacement from the $\\kR$ configurations in the pool.\n\n\\item Steps 2 and 3 are repeated until $\\ceiling$ is equal to the ground state energy of the system.\n\n\\end{enumerate}\n\nDuring the annealing step with ceiling energy $\\ceiling$, the fraction $\\epsilon(\\ceiling)$ of replicas in the pool with energies greater than $\\ceiling^\\prime$ is called the ``culling fraction'' and is used to estimate entropies.\n\nThe equilibrium microcanonical annealing framework described above can be easily generalized to an equilibrium canonical annealing framework with the following two modifications. First, the MCMC procedure must be chosen to equilibrate to a fixed temperature rather than a fixed energy ceiling. Second, the resampling step must select from the configuration pool with Boltzmann reweighting, $e^{-(\\beta^\\prime - \\beta)E_i}$, where $E_i$ is the energy of configuration $i$ and $\\beta^\\prime$ is the inverse temperature succeeding $\\beta$ in the annealing schedule. \n\n\\subsection{\\label{sec:entropy}Entropy estimators}\n\n\nThere are two relevant entropies associated with microcanonical ensemble. The energy ceiling or ``volume'' entropy, $S^c_\\ceiling$ is defined as \n\\begin{equation} \n S^c_\\ceiling = \\log{\\Sigma(\\ceiling)},\n\\end{equation}\nwhere $\\Sigma(\\ceiling)$ is the number of configurations of the system with energy less than or equal to $\\ceiling$. If $\\ceiling'$ is the energy following $\\ceiling$ in the annealing schedule (here the energy spectrum, $\\mce$) and if $\\epsilon(\\ceiling)$ is the culling fraction at energy $\\ceiling$, then a recursive estimator for the ceiling entropy is\n\\begin{equation}\n \\Sec_{\\ceiling'} = \\Sec_\\ceiling + \\log(1-\\epsilon(\\ce)),\n \\label{eq:Sec}\n\\end{equation}\nso that, \n\\begin{equation}\n \\Sec_{\\ce} = \\sum_{\\ce_1 > \\ce} \\log(1-\\epsilon(\\ce_1)) + S^c_\\infty,\n \\label{eq:Secroll}\n\\end{equation}\nwhere the summation is over all energies in the annealing schedule greater than $\\ce$ and $S^c_\\infty$ is the logarithm of the total number of system states.\nThe standard definition of entropy in the microcanonical ensemble, $S_\\ce$, sometimes referred to as the surface entropy, is given by \n\\begin{equation}\n\\label{eq:Smc0}\n\\begin{aligned}\nS_\\ce &=\\log[\\Sigma(\\ceiling) - \\Sigma(\\ceiling')],\\\\\n& \\approx \\log\\left[\\Sigma(\\ceiling)\\epsilon(\\ceiling)\\right],\n\\end{aligned}\n\\end{equation}\nso that the estimator for this entropy is related to the ceiling entropy estimator by,\n\\begin{equation}\n\\label{eq:Smc}\n \\Smc_\\ceiling= \\Sec_\\ceiling + \\log\\epsilon(\\ceiling). \n\\end{equation}\nIn the thermodynamic limit the extensive parts of the two entropies are the same. In what follows we refer to $\\Smc_\\ceiling$ as the ``microcanonical entropy.\"\n\n\\subsection{\\label{sec:level2}Estimators for microcanonical observables}\n\nSuppose $A$ is an observable of the system such as the energy or magnetization. An estimator $\\tilde{A}^{\\rm c}_\\ceiling$ for the equilibrium value of $A$ in the energy ceiling ensemble is\n\\begin{equation}\n\\tilde{A}^{\\rm c}_\\ceiling = \\frac{1}{\\kR}\\sum_{j=1}^{\\kR} A^j\n\\end{equation}\nwhere $A^j$ refers to the value of $A$ in the $j$th replica in the configuration pool for energy ceiling $\\ceiling$. We will generally use a superscript $c$ to indicate a quantity in the energy ceiling ensemble.\n\nAn estimator $\\tilde{A}_\\ce$ of the observable in the microcanonical ensemble at energy $\\ceiling$ is obtained by averaging over only those replicas in the configuration pool that have energies at the ceiling, \n\\begin{equation}\n\\tilde{A}_\\ceiling = \\frac{1}{\\epsilon(\\ce) \\kR}\\sum_{j=1}^{\\kR} A^j \\delta (E^j , \\ceiling)\n\\end{equation}\nwhere $\\delta$ denotes Kronecker function and $E^j$ is the energy of replica $j$. \n\n\\subsection{\\label{sec:level3} Estimators for canonical observables}\n\nThe microcanonical entropy can be used to compute an observable in the canonical ensemble from the observable measured in microcanonical annealing. An estimator of the canonical partition function at inverse temperature $\\beta_c$ is given by\n\\begin{equation}\n\\label{eq:Z}\n\\tilde{Z}(\\beta)=\\sum_{\\ce} e^{-\\beta \\ceiling + \\Smc_\\ceiling}\n\\end{equation}\nwhere the sum is taken over all the energy levels of the system. The canonical energy distribution,\n$\\rho_\\beta(\\ceiling)$ is \n\\begin{equation}\n\\rho_\\beta(\\ceiling)=\\frac{e^{-\\beta \\ceiling + \\Smc_\\ceiling}}{\\tilde{Z}(\\beta)}.\n\\label{eq:canonical_histogram}\n\\end{equation} \nAn estimator $\\tilde{A}(\\beta)$ of the observable $A$ in the canonical ensemble at inverse temperature $\\beta$ is given by\n\\begin{equation}\n\\tilde{A}(\\beta)=\\sum_\\ce \\rho_\\beta(\\ce)\\tilde{A}_{\\ce},\n\\label{eq:MC_to_canonical}\n\\end{equation} \nand an estimator of the free energy is obtained from the definition,\n\\begin{equation}\n\\label{eq:F}\n\\beta \\tilde{F} = - \\log \\tilde{Z}(\\beta).\n\\end{equation}\n\nFor systems which undergo thermal first-order phase transitions, phase coexistence is manifested as a two-peaked canonical energy distribution $\\rho_{\\beta_c} (E)$ at the transition transition temperature, $\\beta_c$. In the thermodynamic limit, the two peaks are delta functions at the energies $E_\\dis$ and $E_\\ord$ of the coexisting disordered and ordered phases, respectively. For an observable $A$, estimators of the coexisting ordered and disordered values at the transition, $\\tilde{A}_\\ord$ and $\\tilde{A}_\\dis$ are, respectively,\n\n\\begin{subequations}\n\\begin{align}\n\\tilde{A}_\\ord &= \\frac{\\sum_{\\ceiling < \\ce_\\od} \\rho_{\\beta_c}(\\ce)\\tilde{A}_{\\ce}}{\\sum_{\\ceiling < \\ce_\\od} \\rho_{\\beta_c}(\\ce)} \\\\[5pt]\n \\tilde{A}_\\dis &= \\frac{\\sum_{\\ceiling \\geq \\ce_\\od} \\rho_{\\beta_c}(\\ce)\\tilde{A}_{\\ce}}{\\sum_{\\ceiling \\geq \\ce_\\od} \\rho_{\\beta_c}(\\ce)},\n\\end{align}\n\\label{eq:peakObs}%\n\\end{subequations}\nwhere the breakpoint energy, $\\ce_\\od$ must be chosen in the range $\\ce_\\ord < \\ce_\\od < \\ce_\\dis$. In the thermodynamic limit, the breakpoint energy can be chosen arbitrarily in this range. For finite systems, an appropriate choice of $\\ce_\\od$ will help minimize finite-size corrections. Clearly, $\\ce_\\od$ should be chosen in the middle of the range to avoid significant overlap with the broadened peaks in the energy distribution. \n\n\\subsection{\\label{sec:weight} Weighted averages}\nIt has been shown \\cite{Mac10a,WaMaKa15b} that for population annealing in the canonical ensemble, multiple independent runs can be combined using weighted averaging to reduce both statistical and systematic errors. In the case of the canonical ensemble the weight factor is proportional to the exponential of the free energy estimator for each run. A similar result holds for equilibrium microcanonical annealing algorithms except that the weight factor is here proportional to the exponential of the entropy estimator \\cite{CaMa17}.\n\nConsider a collection of $M$ independent microcanonical annealing runs, each with the same annealing schedule and the same population parameters $R$ and $\\kR$. Let $\\tilde{A}_{\\ceiling,m}$ denote the average value of an observable $A$ in the microcanonical ensemble from the $m$th run. The best estimate $\\bar{A}_\\ce$ from the collection of $M$ runs is given by the weighted average, \n\\begin{equation}\n\\bar{A}_\\ce = \\sum_{m=1}^{M}\\omega_{\\ce,m} \\tilde{A}_{\\ceiling,m} ,\n\\label{eq:reweighting}\n\\end{equation} \nwhere the weights are proportional to the exponential of the entropy estimator $\\Smc_{\\ceiling,m}$ of run $m$,\n\\begin{equation}\n\\label{eq:weight}\n\\omega_{\\ce,m}=\\frac{e^{\\Smc_{\\ceiling,m}}}{\\sum_{m'=1}^{M} e^{\\Smc_{\\ceiling,m'}}}.\n\\end{equation}\nAn analogous result holds for the energy ceiling ensemble, \n\\begin{equation}\n\\bar{A}_\\ce^c = \\sum_{m=1}^{M}\\omega^c_{\\ce,m} \\tilde{A}^c_{\\ceiling,m} ,\n\\label{eq:reweightingc}\n\\end{equation}\nwhere, \n\\begin{equation}\n\\omega^c_{\\ce,m}=\\frac{e^{\\Sec_{\\ceiling,m}}}{\\sum_{m'=1}^{M} e^{\\Sec_{\\ceiling,m'}}}.\n\\label{eq:weightc}\n\\end{equation}\nFor either ensemble, if a run fails at energy $\\ce$ then for all energies $\\ce^\\prime \\geq \\ce$ the corresponding weight vanishes.\n\nIt is most straightforward to first derive the weighted average formula for the energy ceiling result and then the microcanonical result. Consider the simple situation that at annealing step $\\ce$ the weights for all runs are the same, $\\omega^c_{\\ce,m} \\equiv 1\/M$. At the next annealing step with energy ceiling $\\ce^\\prime$, the population to be averaged over in run $m$ has a size $R\\kR(1-\\epsilon_{m}(\\ce))$ where $\\epsilon_{m}(\\ce)$ is the culling fraction at energy ceiling $\\ce$ in run $m$. An unbiased average in the energy ceiling ensemble over many runs at energy ceiling $\\ce^\\prime$ should weight each run according to the size of its population, which requires that run $m$ be weighted by a factor proportional to $(1-\\epsilon_{m}(\\ce))$. From Eq.\\ \\eqref{eq:Sec} we have that $(1-\\epsilon_{m}(\\ce))$ is the exponential of the volume entropy change from $\\ce$ to $\\ce^\\prime$. \n\nIn the more general case where runs already have differing weights at energy ceiling $\\ce$ we have the recursion relation,\n\\begin{equation}\n\\omega^c_{\\ce^\\prime,m} \\propto (1-\\epsilon_m(\\ce))\\omega^c_{\\ce,m}, \n\\end{equation}\nwhere the constant of proportionality is independent of run and is set by the normalization of the weights.\nFrom Eq.\\ \\eqref{eq:Sec} we see that $(1-\\epsilon_{m}(\\ce))$ is the exponential of the volume entropy change from $\\ce$ to $\\ce^\\prime$. Collapsing the telescoping product of entropy changes yields the weight factor given in Eq.\\ \\eqref{eq:weightc}.\n\nTo obtain the analogous result for the microcanonical ensemble, note that the population that is averaged in the microcanonical ensemble is a factor $\\epsilon_m(\\ce)$ smaller than the population averaged in the ceiling ensemble. Thus the microcanonical ensemble and ceiling ensemble weights are related by, \n\\begin{equation}\n \\omega_{\\ce,m} \\propto \\epsilon_m(\\ce)\\omega^c_{\\ce,m},\n\\end{equation}\nand from Eq.\\ \\eqref{eq:Smc} we see that this transformation is precisely the transformation from $\\Sec$ to $\\Smc$, thus verifying \\eqref{eq:weight}.\n\nFormulas for the weighted average of the two entropies are slightly more complicated to derive because the entropy depends on a thermodynamic integration over all of the annealing steps, Eq.\\ \\eqref{eq:Secroll}. Thus the weighted average volume entropy can be written as,\n\\begin{equation}\n \\bar{S}^c_\\ce = \\log \\prod_{\\ce_1 > \\ce} \\, \\sum_{m=1}^M (1 - \\epsilon_{m}(\\ce_1))\\omega^c_{\\ce_1,m} + \\Sec_\\infty.\n\\end{equation}\nExpanding the definition \\eqref{eq:weightc} of the weight factor and using Eq.\\ \\eqref{eq:Sec} to relate the culling factor to the change in entropy yields the telescoping product,\n\\begin{equation}\n \\bar{S}^c_\\ce = \\log \\prod_{\\ce_1 > \\ce} \\frac{\\sum_{m=1}^M \\exp(\\Sec_{\\ce_1^\\prime,m})}{\\sum_{m=1}^M \\exp(\\Sec_{\\ce_1,m})} + \\Sec_\\infty,\n\\end{equation}\nwhere $\\ce_1^\\prime$ is the successor to $\\ce_1$ in the annealing schedule. After canceling terms in the numerator and denominator we have,\n\\begin{equation}\n \\bar{S}^c_\\ceiling = \\log \\frac{1}{M}\\sum_{m=1}^{M}e^{\\Sec_{\\ceiling,m}},\n \\label{eq:reweighting_S0}\n\\end{equation} \nand, analogously for the microcanonical entropy, we have\n\\begin{equation}\n \\bar{S}_\\ceiling = \\log \\frac{1}{M}\\sum_{m=1}^{M}e^{\\Smc_{\\ceiling,m}}.\n \\label{eq:reweighting_S}\n\\end{equation} \n\nWe emphasize that for fixed $R$ and $\\kR$, results from weighted averaging are exact in the limit $M\\rightarrow \\infty$ for any fixed number of MC sweeps per run. \n\nWeighted averaging in the canonical ensemble is discussed in Refs.\\ \\cite{Mac10a,WaMaKa15b}. Given $M$ independent simulations in the canonical ensemble, the best estimate $\\bar{A}(\\beta)$ of an observable $A$ in the canonical ensemble at inverse temperature $\\beta$, is given by\n\\begin{equation}\n \\bar{A}(\\beta)= \\frac{ \\sum_{m=1}^M \\tilde{A}_m(\\beta) e^{-\\beta \\tilde{F}_m}}{\\sum_{m=1}^M e^{-\\beta \\tilde{F}_m}}\n \\label{eq:free_energy_rw}\n\\end{equation} \nwhere $\\beta \\tilde{F}_m = -\\log \\tilde{Z}_m(\\beta)$ is the free energy estimator of the $m$th run. In analogy to Eq.\\ \\eqref{eq:reweighting_S}, the best estimator of the free energy from multiple runs is given by,\n\\begin{equation}\n-\\beta \\bar{F} = \\log \\frac{1}{M}\\sum_{m=1}^{M}e^{-\\beta \\tilde{F}_m}.\n\\label{eq:reweighting_F}\n\\end{equation} \n\nSince our simulations directly produce results in the microcanonical ensemble, it is important to show that transforming from the microcanonical to the canonical ensemble commutes with carrying out a weighted average in either ensemble. Specifically, we would like to show that, \n\\begin{equation}\n \\label{eq:commute_A}\n \\bar{A}(\\beta)=\\frac{ \\sum_{\\ce}\\bar{A}_\\ce \\ e^{-\\beta \\ce + \\bar{S}_E} }{\\sum_{\\ce} e^{-\\beta \\ce + \\bar{S}_\\ce}},\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:commute_F}\n -\\beta \\bar{F}=\\log \\sum_{\\ce} e^{-\\beta \\ce + \\bar{S}_\\ce}, \n\\end{equation}\nwhere the canonical weighted averages, $\\bar{A}(\\beta)$ and $\\beta \\bar{F}$ are defined in \\eqref{eq:free_energy_rw} and \\eqref{eq:reweighting_F}, respectively, and where on the RHS the barred quantities are microcanonical weighted averages defined in, Eqs.\\ \\eqref{eq:reweighting}, \\eqref{eq:weight} and \\eqref{eq:reweighting_S}. To verify \\eqref{eq:commute_F}, simply substitute the definition \\eqref{eq:reweighting_S} of $\\bar{S}_\\ce$ on the RHS, exchange the order of summation and then invoke the definitions, \\eqref{eq:Z} and \\eqref{eq:F}. A similar but slightly more involved calculation verifies Eq.\\ \\eqref{eq:commute_A}. Finally, we note that the weighted averages for observables, $A_\\ord$ and $A_\\dis$, defined in Eqs.\\ \\eqref{eq:peakObs} are obtained from Eq.\\ \\eqref{eq:commute_A} with the summation over a restricted range of energies, $\\ce<\\ce_\\od$ and $\\ce \\geq \\ce_\\od$, respectively. Alternatively, results analogous to Eq.\\ \\eqref{eq:free_energy_rw} for $A_\\ord$ and $A_\\dis$ are obtained by replacing $\\tilde{F}$ by $\\tilde{F}_\\ord$ and $\\tilde{F}_\\dis$, respectivly, where these partial free energies are defined in the obvious way by limiting the range of summation in the partition function, Eq.\\ \\eqref{eq:Z}.\n\n\\subsection{Systematic errors}\n\\label{sec:errors}\nIt is possible to quantify systematic errors in microcanonical annealing algorithms from the run-to-run variance of the entropy estimator. This result follows from the fact that weighted averages are exact in the limit $M \\rightarrow \\infty$. Systematic errors are then given by the difference between weighted and unweighted averages in this limit. The result, derived for the case of population annealing ($R\\gg 1$ and $\\kR = R$) in Refs.\\ \\cite{WaMaKa15b,CaMa17}, is that the systematic error, $\\Delta A_\\ce$ in the estimator $\\tilde{A}_\\ce$ is given by \nthe covariance of $\\tilde{A}_\\ce$ and $\\Smc$,\n\\begin{equation}\n \\label{eq:syserror}\n \\Delta A_\\ce = {\\rm cov}(\\tilde{A}_\\ce,\\Smc) = {\\rm var}(\\Smc) \\left[\\frac{{\\rm cov}(\\tilde{A}_\\ce,\\Smc)}{{\\rm var}(\\Smc)}\\right].\n\\end{equation}\nThe second, trivial identity in Eq.\\ \\eqref{eq:syserror} is useful because the ratio in the square brackets goes to a constant that depends on the observable $A$ as the simulation size, $\\kR$ becomes large, while ${\\rm var}(\\Smc)$ is expected vanish as $1\/\\kR$ since the number of measurements going into estimating $\\Smc$, also becomes large. Note that the scaling of the simulation size depends on the type of annealing algorithm: in equilibrium simulated annealing $\\kR$ becomes large with $R=1$ and in population annealing, $\\kR=R$, while various options are possible for hybrid annealing. In any case, systematic errors and their scaling with simulation size are characterized by the behavior of ${\\rm var}(\\Smc)$. Although this result was originally derived for population annealing, it holds for the whole class of equilibrium microcanonical annealing protocols discussed here since it depends on only the the validity of weighted averaging. An analogous result holds for quantities in the energy ceiling ensemble. \n\nThe systematic error of an observable in the canonical ensemble, $\\Delta A(\\beta)$ is given by,\n\\begin{equation}\n \\label{eq:syserrorcan}\n \\Delta A(\\beta) = {\\rm cov}(\\tilde{A}(\\beta),\\beta \\tilde{F}) = {\\rm var}(\\beta \\tilde{F}) \\left[\\frac{{\\rm cov}(\\tilde{A}(\\beta),\\beta \\tilde{F})}{{\\rm var}(\\beta \\tilde{F})}\\right].\n\\end{equation}\nAfter transforming from the microcanonical to the canonical ensemble, this result can be applied to quantify errors in canonical observables obtained from microcanonical simulations. Results for systematic errors for $A_\\ord$ and $A_\\dis$ can be obtained from the variance of the partial free energies $\\tilde{F}_\\ord$ by $\\tilde{F}_\\dis$, respectively, discussed at the end of Sec.\\ \\ref{sec:weight}.\n\nThe variance of entropy and free energy estimators are therefore a useful tool for comparing the performance of different versions of equilibrium microcanonical annealing.\n\n\\subsection{Relation to other annealing algorithms}\n\nThe case $R=1$ and $\\kR \\gg 1$ is a form of simulated annealing. Simulated annealing is conventionally carried out in the canonical ensemble with the last configuration generated by the MCMC procedure used as the starting point for the next annealing step. Thus, conventional simulated annealing is the case $R=\\kR=1$. If this conventional choice were applied in the microcanonical ensemble the algorithm would fail with probability $\\epsilon$ at each annealing step and would be highly inefficient. The case $R \\gg 1$ and $\\kR=R$ is microcanonical population annealing, which has previously been applied to study hard sphere systems \\cite{CaMa17}. We refer to the intermediate cases $1< R < \\kR$ as hybrid annealing. In what follows we abbreviate population annealing, simulated annealing and hybrid annealing as PA, SA and HA, respectively.\n\nOne of our goals is to understand the trade-offs related to the population size $R$. One conclusion that seems evident is that a single large run of a hybrid algorithm with parameters $R=R_{\\rm h}$ and $\\kR=\\kR_{\\rm h}$ should outperform the weighted average of $M=R_{\\rm h}$ runs of simulated annealing with $\\kR=\\kR_{\\rm h}\/M$ even though both simulations require nearly the same amount of computational work. The argument is that resampling from the larger pool in the hybrid algorithm will be more effective than simply reweighting independent runs. A related result was observed in comparing standard canonical simulated annealing with canonical population annealing for finding ground states of spin glasses~\\cite{WaMaKa15}. \n\n\\section{\\label{sec:model_and_observables}Model and Observables}\n\\subsection{\\label{sec:model}The $q$-state Potts Model}\nThe $q$-state Potts model \\cite{Wu82} is a generalization of the Ising model consisting of $N$ interacting spins $\\{s_i|i=1, \\ldots, N\\}$ each taking values $s_i \\in \\{1,...q\\}$. The energy, $\\ce$ is given by,\n\\begin{equation}\n \\ce=-\\sum_{ \\langle i,j \\rangle} \\delta(s_i,s_j), \n\\end{equation}\nwhere $\\delta$ is the Kronecker delta function and the summation is over all the interacting pairs of spins. For spins on a square (two-dimensional) lattice with nearest neighbor interactions and for $q>4$, the Potts model undergoes a thermal first-order phase transition at inverse temperature $\\beta_c = \\log(1+\\sqrt{q})$. Higher $q$-values result in stronger first-order phase transitions. In our simulations we consider the two-dimensional, $20$-state Potts model on an $L\\times L$ square lattice with periodic boundary conditions. \n\nThe first-order phase transition of the Potts model in the canonical ensemble is between a disordered phase with magnetization zero and energy per spin $e_\\dis$ and a magnetically ordered phase with magnetization per spin $m_\\ord$ and energy per spin $e_\\ord$. Configurations with energy per spin between these values possess an interface between ordered and disordered domains, leading to a free energy barrier proportional to the system size $L$. \n\nIn the microcanonical ensemble configurations with an interface are stable for energies per spin between $e_\\dis$ and $e_\\ord$ making it possible for an annealing algorithm to traverse the coexistence region nearly continuously. Nonetheless, there is a series of four discontinuous transitions in the microcanonical ensemble, which introduce barriers between the disordered and ordered phases and increase equilibration times. First, at a length scale dependent energy, $e_1(L)e_\\ord$, there is an evaporation transition where the disordered droplet vanishes, leaving the homogeneous ordered phase. Mirroring $e_1(L)$, $e_4(L)$ depends on system size and approaches $e_\\ord$ from above. \n\n\n\\subsection{Observables \\label{sec:observables}}\n\nIn this section we define the observables measured in our simulations. First, we measure the entropy using Eq.\\ \\eqref{eq:Smc} and use it to calculate the free energy and the canonical energy distribution of Eq.\\ \\eqref{eq:canonical_histogram} at the transition point $\\beta_c$. From these quantities we calculate the following observables.\n\n\\paragraph*{Energies of ordered and disordered phase at coexistence}\n Using the canonical energy distribution and Eqs.\\ \\eqref{eq:peakObs} we measure the ordered and disordered phase energies per spin, $e_\\ord$ and $e_\\dis$, respectively. \n\n \\paragraph*{Peak ratio and disordered phase excess}\n We measure two observables related to the relative weights of the ordered and disordered peaks in the energy distribution at $\\beta_c$, the peak ratio and the disordered phase excess, both of which are defined below. \n \n The peak ratio $\\rc$ at the transition temperature is defined as,\n\\begin{equation}\n \\rc = \\frac{\\sum_{\\ceiling < \\ce_\\od} \\rho_{\\beta_c}(\\ce)} \n {\\sum_{\\ce \\geq \\ce_\\od} \\rho_{\\beta_c}(\\ce)}.\n \\label{peakRatio}\n\\end{equation}\nThe peak ratio is related to the difference in the free energies of the ordered and disordered phases, which in the thermodynamic limit are equal at the transition temperature. However, since there are $q$ different ordered phases, the exact value of the peak ratio in the thermodynamic limit is $q$ at the transition temperature.\n\nSince the peak ratio is not an observable of the form of Eq.\\ \\eqref{eq:MC_to_canonical}, the weighted average cannot be computed simply from Eq.\\ \\eqref{eq:free_energy_rw}. We define another observable related to the peak ratio called the disordered phase excess $\\A$, \n\\begin{equation}\n \\A = \\sum_{\\ceiling > \\ce_\\od} \\rho_{\\beta_c}(\\ce) - \\frac{1}{q + 1},\n \\label{eq:disordered_excess}\n\\end{equation}\nwhere the subtracted constant is chosen so the exact value of $\\A$ in the thermodynamic limit is zero. The disordered phase excess is used as a measure of accuracy of a simulation at the transition point. Unlike the peak ratio, a weighted average of $\\A$ using Eq.\\ \\eqref{eq:free_energy_rw} gives the best estimate of $\\A$ and is exact in the limit $M \\rightarrow \\infty$.\n\n\\paragraph*{Wrapping fraction} To estimate the locations $e_2$ and $e_3$ of the droplet\/stripe transitions we define the wrapping fraction, $\\omega_w$, as the average number of directions wrapped by connected paths of like spins. The number of wrapping directions takes values in $\\left\\{ 0,1,2\\right\\}$; in the disordered phase and the droplet state the wrapping number is zero, in the stripe state the wrapping number is one, while in the ordered phase the wrapping number is two. \n\n\\paragraph*{Droplet fraction} In order to detect the evaporation transition between a configuration containing a disordered droplet and a homogeneous ordered phase, we consider the disordered cluster size histogram $\\csh$, where here a disordered cluster is defined using the following procedure. First, identify the largest connected cluster of adjacent spins having the same value. Near the evaporation transition point this cluster will wrap both vertically and horizontally. Label all spins in this cluster $``0\"$ and label all other spins $``1\"$. Group adjacent spins labelled $``1\"$ into clusters, and let $\\csh$ be the configuration averaged histogram of the sizes of these disordered clusters.\n\nThe disordered cluster size distribution is shown in Fig.\\ \\ref{fig:csh} for three energies. At energies lower than the evaporation transition point $e_4 (L)$, all clusters consist of excitations within a homogeneous ordered phase, and $\\csh$ decays exponentially. At the transition point $\\csh$ is observed to be a power law distribution as the disordered droplet breaks down into smaller disordered regions. At energies greater than the transition point, a finite fraction of spins exist within a disordered droplet, and thus, in addition to a power law, $\\csh$ contains a peak at some value greater than zero. \n\nIn order to measure the droplet fraction $\\omega_c$, which we define as the probability that a configuration contains a large droplet of disordered phase, we subtract the power law component from $\\csh$ and integrate over the remaining peak. In practice, this is done by finding the least squares fit of $\\csh$ to a power law up to a cutoff size that is chosen so that the fit only covers the power law region of $\\csh$. What remains is a single peak centered on the characteristic droplet size for the given energy. \n\nWhen $\\csh$ is averaged over many measurements, we expect the integral of this peak to be one for energies sufficiently above the transition point. In other words, we expect every configuration to contain one droplet that is significantly larger than the droplet size distribution predicted by the power law. As the energy approaches $e_4$ from above, this peak will decrease in size and $\\omega_c$ will drop to zero as the power law behavior of $\\csh$ crosses over to exponential decay. In this regime our procedure becomes inaccurate as it becomes more difficult to fit $\\csh$ to a power law, and in general this method of measuring $\\omega_c$ is only valid near the transition point. \n\n\\paragraph*{Magnetization integrated autocorrelation time}\n\nThe magnetization per spin, $m$ for the Potts model is defined as, \n\\begin{equation}\nm= \\frac{q(N_{\\rm max}\/N) - 1}{q-1},\n\\end{equation}\nwhere $N_{\\rm max}$ is the maximum number of spins taking the same value.\nIn the disordered phase $N_{\\rm max}\\approx N\/q$ and the magnetization vanishes, while in the ordered phase $N_{\\rm max} > N\/q$. As the temperature approaches zero, $N_{\\rm max} \\rightarrow N$ and the magnetization per spin approaches one.\n\nThe integrated autocorrelation time $\\tau$ of the magnetization is defined by\n\\begin{equation}\n \\tau = \\frac{1}{2} + \\frac{\\sum_{t=1}^{\\infty} ( \\langle m(t_0)m(t_0 + t)\\rangle-\\langle m \\rangle ^2)}{\\langle m^2 \\rangle -\\langle m \\rangle ^2}\n \\label{eq:autocorrelation}\n\\end{equation} \nwhere time is measured in Monte Carlo sweeps and the brackets refer either to an ensemble average or an average over starting times $t_0$. The integrated autocorrelation time sets the time between independent measurements. Although the asymptotic decay of the correlation functions, i.e.\\ the exponential autocorrelation time, is a better measure of the time to reach equilibrium, $\\tau$ is usually also a good approximate measure of the equilibration time or, equivalently, the computational work required to equilibrate the system at a given energy.\n\nThe time series used to compute $\\tau$ must be much larger than $\\tau$ itself and and upper limit of the sum in Eq.\\ \\eqref{eq:autocorrelation} must be finite. Here we use an iterative method described in \\cite{OsSo04} that minimizes systematic errors and is used to set the upper limit of the sum defining $\\tau$.\n\n\n\\section{Simulation details}\n\\label{sec:simdet}\nWe perform SA, HA, and PA simulations for the 2D 20-state Potts model for systems sizes $L = 30,40,50,$ and $60$. For each size, the computational work, measured in sweeps, is kept the same for each algorithm. In each case, we obtain results by using weighted averaging over $M$ simulation runs. The number of MC sweeps performed at each energy is equal to $n_s$ given by\n\\begin{equation}\n n_s(\\ce)=\n \\begin{cases}\n \\as,& \\text{if } \\ce > -\\frac{N}{2}\\\\\n 20 \\as,& \\text{if } -\\frac{N}{2} \\geq \\ce > -\\frac{3N}{2}\\\\\n 5 \\as,& \\text{if } -\\frac{3N}{2} > \\ce \n \\end{cases}\n \\label{eq:sweep_schedule}\n\\end{equation}\nwhere $\\as$ is a parameter that depends on the algorithm and is chosen to make the computational work roughly equal between algorithms for the same system size. \n\nThis {\\it ad hoc} sweep schedule concentrates the majority of MC sweeps in the range of energies where the integrated autocorrelation time is highest. We have found that this sweep schedule \nis an improvement compared to a uniform sweep schedule, but is not claimed to be optimal.\n\nSimulation parameters for the three annealing algorithms are presented in Table \\ref{table:simulation_parameters}. Since both the number of proposed updates per MC sweep and the number of annealing steps in the simulation grow linearly with the system size $N$, the work per simulation grows quadratically for a given sweep parameter $\\as$. In order to keep the run times manageable, fewer MC sweeps ($\\as$ decreased) are performed per annealing step for larger system sizes. In order to partially compensate for the decrease in the number of sweeps, reweighted averages are performed over more runs ($M$ increased) for larger system sizes. \n\nIt is straightforward to take advantage of the massively parallel nature of the HA and PA algorithms by performing MC sweeps on independent replicas simultaneously. As the majority of computational work in these simulations is spent performing MC sweeps and only a small fraction is spent during the resampling step, significant improvement in wall clock time can be gained for PA and HA through parallelization despite doing the resampling sequentially. \n\nFor PA the speedup due to parallelization is most dramatic, because the number of independent replicas is on the order of $10^5$. Population annealing simulations were implemented on a single NVIDIA Tesla C2075 GPU using the $CUDA$ library using the methods described in Ref.\\ \\cite{BaWeBo17}. One thread was assigned to each replica so that the total number of threads was equal to the population size. Despite the simplicity of this implementation, we still observed a notable speed-up with PA when compared to SA with the same number of sweeps running on a CPU. For instance, for our $L=50$ simulations, the PA runs were about $85$ times faster than SA runs.\n\nFor our HA simulations the population size of $100$ was not large enough to benefit from using a GPU though a significant speedup over SA was obtained using OpenMP with $16$ threads.\n\n\t\\begin{table}\n\t\t\\begin{tabular}{ |c|c|c|c|c|c|} \n\t\t\t\\hline\n\t\t\tAlgorithm & $R$ & $L$ & $\\as$ & $\\kR\/R$ &$M$\\\\\n\t\t\t\\hline\n \\multirow{4}{*}{SA} & \\multirow{4}{*}{1} & 30 &\t$1.2 \\times 10^6$ & $1.2 \\times 10^6$ & 30\t\\\\ \t\n & \t\t\t\t\t & 40 &\t$3.8 \\times 10^5$ & $3.8 \\times 10^5$ & 40 \\\\ \n & \t\t\t\t\t & 50 &\t$1.6 \\times 10^5$ & $1.6 \\times 10^5$ & 50\t \\\\\n & \t\t\t\t\t & 60 &\t$7.5 \\times 10^4$ &\t$7.5 \\times 10^4$ & 60\t \\\\\n \\hline\n \n \\multirow{4}{*}{HA} & \\multirow{4}{*}{100} & 30 &\t$1.2 \\times 10^4$ &$1.2 \\times 10^3$& 30 \\\\ \n & \t\t\t\t\t & 40 &\t$3.8 \\times 10^3$ &$3.8 \\times 10^2$& 40 \\\\ \n & \t\t\t\t\t& 50 & $1.6 \\times 10^3$ & $1.6 \\times 10^2$ & 50 \\\\\n & \t\t\t\t\t\t& 60 & $7.5 \\times 10^2$ & $7.5 \\times 10$ & 60 \\\\\n\n \\hline\n \\multirow{4}{*}{PA} & \t$8 \\times 10^5$\t & 30 &\t2\t& 1 \t& 30\t \\\\ \n &\t$2 \\times 10^5$ \t& 40 &\t1.9\t& 1\t& 40\t \\\\ \n &\t$10^5$ \t\t& 50 & 1.6& 1\t& 50\t \\\\\n &\t$7.5 \\times 10^4$ & 60 &\t1\t& 1\t& 60\t \\\\\n \\hline\n \n\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\caption{Simulation parameters for SA, HA, and PA. \n Values in the table are system size $L$, population size $R$, the parameter $\\as$ setting the number of MCMC sweeps (see Eq.\\ \\eqref{eq:sweep_schedule}), number of runs $M$ used for weighted averaging, and ratio of pool size to population size, $\\kR\/R$.\n }\n\\label{table:simulation_parameters}\n\\end{table}\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{Plots\/snapshots.png}\n \\caption{Equilibrium configurations of the 20-state Potts model with $L=40$ and periodic boundary conditions. From left to right and then bottom to top configurations are shown in the (1) homogeneous disordered phase, (2) disordered phase with an ordered droplet, (3) stripe phase, (4) ordered phase with disordered droplet, and (5) homogeneous ordered phase.}\n \\label{fig:snapshots}\n\\end{figure}\n\n\\begin{figure}\n \\hspace{-12pt}\n \\begin{tikzpicture}[yscale=.9, xscale = .9]\n \\pgfplotsset{every axis\/.append style={\n xlabel={$e$},\n ylabel={$\\rho(e)$},\n ymin=10e-18,\n ymax = 10e2\n },\n }\n \\begin{axis}[ymode=log, legend pos= south east, execute at begin axis={\n \\draw (rel axis cs:0,0) -- (rel axis cs:1,0)\n (rel axis cs:0,1) -- (rel axis cs:1,1);\n }]\n \\addplot[red] table [mark=none, x=e, y=pe, col sep=space] {Plots\/PA_hist_900.tex};\n \\addlegendentry{L=30}\n \\addplot[blue] table [mark=none, x=e, y=pe, col sep=space] {Plots\/PA_hist_1600.tex};\n \t\\addlegendentry{L=40}\n \\addplot[green] table [mark=none, color = green, x=e, y=pe, col sep=space] {Plots\/PA_hist_2500.tex};\n \t\\addlegendentry{L=50}\n \\addplot[black] table [mark=none, x=e, y=pe, col sep=space] {Plots\/PA_hist_3600.tex};\n \t\\addlegendentry{L=60}\n \\end{axis}\n \\end{tikzpicture}\n \\caption{Distribution $\\rho_\\beta(e)$ of the energy per spin, $e$ at the transition temperature calculated with population annealing for system sizes $L =$ 30, 40, 50, and 60 (from top to bottom).}\n \\label{fig:canonical_histogram}\n\\end{figure}\n\n\\section{\\label{sec:Results}Results}\nIn this section we describe the results of our simulations. Our primary objective is comparing the three annealing algorithms but we also obtain precise finite-size results for the Potts model observables described above.\n\nIn order to obtain results in the canonical ensemble at the transition temperature, the microcanonical entropy is measured at every energy using each of the microcanonical annealing algorithms and the results are reweighted to obtain the free energy at the transition using Eqs.\\ \\eqref{eq:Z} and \\eqref{eq:F}. The peak ratio observables, $\\rc$ and $\\A$ are measured using Eqs.\\ \\eqref{eq:peakObs} and \\eqref{eq:disordered_excess}, respectively. Ordered and disordered phase energies per spin, $e_\\ord$ and $e_\\dis$, are calculated from the canonical energy histogram, Eq.\\ \\eqref{eq:canonical_histogram} using Eqs.\\ \\eqref{eq:peakObs}. For the system sizes considered, the ordered and disordered peaks in the energy distribution at coexistence, shown in Fig.\\ \\ref{fig:canonical_histogram}, are sufficiently separated that any ambiguity in defining breakpoint energy, $E_\\od$ between ordered and disordered phases is negligible, and in practice $E_\\od$ was chosen to be the point of minimum weight. The reported values of all observables are obtained from a weighted average over $M$ independent simulations as described in Sec.\\ \\ref{sec:weight} and error bars are obtained by bootstrapping over these runs. \n\nTable \\ref{tab:results} shows results from the three algorithms and four system sizes for $e_\\dis$, $e_\\ord$, $\\rc$, $\\A$ and $\\mathrm{var}(\\beta F)$. The results for SA are accurate to four or five significant digits and the results for the different algorithms are in agreement with one another within error bars. The SA results for size $L=40$ are consistent with and more accurate than the results reported in Ref.\\ \\cite{JaKa97}.\n\nBased on both the error bars, which quantify statistical errors, and on $\\mathrm{var}(\\beta F)$, which quantifies systematic errors (see Eq.\\ \\eqref{eq:syserrorcan}, we see that SA yields significantly more accurate and well-equilibrated results than either PA or HA. Furthermore, except for size $L=60$, PA is better than HA. The relative ranking of the three algorithms is also visible in Figs.\\ \\ref{fig:bf_ed_scatter} and \\ref{fig:bf_X_scatter}, which are scatter plots of $e_\\dis$ and $\\A$, respectively, vs.\\ the free energy per spin, $\\beta f$, for size $L=40$. Each point represents one of the $M$ independent runs. The larger the variance of the value of the observable, the larger statistical errors for the observable while a large covariance with $\\beta f$ indicates a large systematic error according to \\eqref{eq:syserrorcan}.\n\nWhile measuring observables at the transition can be done relatively efficiently using the microcanonical ensemble, the simulations become more difficult with increasing system size. It is believed that the major sources of hardness in these simulations are the four microcanonical phase transitions described in Sec.\\ \\ref{sec:model}. To investigate this we used the integrated autocorrelation time of the magnetization $\\tau$ as a measure of computational hardness. \n\nWe plot $\\tau$ in Fig.\\ \\ref{fig:autocorrelation_by_size} as function of the energy ceiling for the four system sizes considered. Indeed, $\\tau$ displays sharp peaks that grow with system size. While only three peaks are visible for the system sizes considered, we believe that the largest, rightmost peak is associated with both the condensation transition and the first wrapping transition. For larger system sizes we expect that this peak would break into two distinct peaks.\n\nThe evaporation transition is studied using methods described in Sec.\\ \\ref{sec:observables}. The disordered cluster size histogram $\\csh$ is plotted in Fig.\\ \\ref{fig:csh} at three energies for size $L=40$. As described in Sec.\\ \\ref{sec:observables} we observe a power law with an additional distinct peak for energies above the evaporation transition, power law behavior at the transition, and exponential decay below. The integral of the peak above the fitted power law in Fig.\\ \\ref{fig:csh} gives a measurement of the $\\omega_c$. The power law behavior of $\\csh$ at the evaporation transition is unexpected and warrants further investigation.\n\nIn Fig.\\ \\ref{fig:autocorrelation} we show the magnetization integrated autocorrelation time, $\\tau$, the wrapping fraction, $\\omega_w$, and droplet fraction, $\\omega_c$, for $L=40$. The wrapping fraction is zero at high energies and displays two jumps as expected in the coexistence region $e_o < e < e_d$. These jumps coincide with the two rightmost peaks in $\\tau$, suggesting that they are in fact points where increased computational effort is needed. \n\nWe observe that $\\omega_c \\rightarrow 1$ for energies far above the condensation transition, and drops quickly to zero below the transition at energy $e_4$. The point where $\\omega_c$ begins to rapidly decrease marks the point where a droplet of disordered phase becomes unstable, which coincides in Fig.\\ \\ref{fig:autocorrelation} with the final peak in $\\tau$. \n\nThe growth of the the autocorrelation time due to the wrapping and condensation\/evaporation transitions poses the greatest challenge to any microcanonical algorithm including microcanonical annealing. These transitions limit the system sizes that can be feasibly studied and ultimately will lead to exponential growth in the resources required to perform simulations. However, in Fig.\\ \\ref{fig:autocorrelation_by_size} we observe only a modest increase in $\\tau$ with increasing system size. Compare this to the canonical energy histogram in Fig.\\ \\ref{fig:canonical_histogram}, where the depth of the valley between ordered and disordered grows exponentially for the same sizes considered. The depth of this valley provides an estimate for the time scale needed by a canonical MCMC algorithm to tunnel between an ordered phase and the coexisting disordered phase. This suggests that for modest system sizes microcanonical annealing is not only more efficient than canonical MCMC but also achieves better scaling until exponential slowing eventually dominates. \n\n\\begin{center}\n\t\\begin{table*}\n\n\t\t\\begin{tabular}{ |c|c|c|c|c|c|c|} \n\t\t\t\\hline\n\t\t\t$L$ & Algorithm & $e_\\dis$ & $e_\\ord$ & $\\rc$& $\\A$ &$\\mathrm{var}(\\beta F)$\\\\\n\t\t\t\\hline\n \\multirow{4}{*}{30} & \t\t SA\t\t & -0.626551(25) &-1.820723(12)&19.965(97)&7.9e-5 $\\pm$ 2.2e-4 &0.0013\\\\\n & \t\t HA \t\t & -0.62678(73) &-1.81966(90)&18.1(1.6)&4.9e-3 $\\pm$ 8.1e-3 &0.38\\\\ \n & \t\t PA\t\t & -0.626603(48) &-1.820732(49)&20.17(29)&-3.9e-4 $\\pm$ 6.6e-4&0.012\\\\\n \\hline\n \\multirow{4}{*}{40} & \t\t SA\t\t & -0.626514(34)&-1.820632(17)&19.84(15)&3.6e-4 $\\pm$ 3.5e-4&0.0049\\\\\n & \t\t HA \t\t & -0.62619(49)&-1.82017(32)&15.0(1.0)&1.5e-2 $\\pm$ 7.1e-3 &0.41\\\\ \n & \t\t PA\t\t & -0.62661(11)&1.82050(11)&19.39(69)&1.4e-3 $\\pm$ 1.8e-3 &0.064\\\\\n \\hline\n \\multirow{4}{*}{50} & \t\t SA\t\t & -0.626525(41)&-1.820649(20)&19.45(31)&1.3e-3 $\\pm$ 7.6e-4 &0.072\\\\ \t\t\n & \t\t HA \t\t & -0.62689(19)&-1.81991(30)&15.1(1.1)&1.4e-2 $\\pm$ 7.8e-3&0.44\\\\\n & \t\t PA\t\t & -0.62663(12)&-1.82072(15)&21.5(1.2)&-3.5e-4 $\\pm$ 4.2e-3&0.25\\\\\n \\hline\n \\multirow{4}{*}{60} & \t\t SA\t\t & -0.626537(41)&-1.820753(31)&19.62(46)&8.7e-4 $\\pm$ 1.1e-3 &0.096\\\\ \t\t\t\t\t\n & \t\t HA \t\t & -0.62652(18)&-1.82050(36)&20.31(91)&-7.0e-4 $\\pm$ 5.8e-3 &0.70\\\\ \n & \t\t PA\t &-0.62660(18)&\t-1.82081(34)& 22.93(83)&-5.8e-3 $\\pm$ 1.7e-2& 1.3\\\\\n \\hline\n $\\infty$ (exact)\t\t\t\t&\t\t\t-\t\t\t&-.626529\\ldots &-1.820684\\ldots&20& 0 &\t\\\\\n \\hline\n\t\t\\end{tabular}\n\t\t\\caption{Energies of the disordered phase, $e_\\dis$, ordered phase, $e_\\ord$, the peak ratio, $\\rc$, disordered phase excess $\\A$, and the variance of the free energy, $\\mathrm{var}(\\beta F)$, at the transition temperature. Results are shown for the SA, HA and PA simulations, and exact values in the thermodynamic limit \\cite{Ba73}.}\n\t\\label{tab:results}\n\t\\end{table*}\n\\end{center}\n\n\\begin{figure}[h]\n \\begin{tikzpicture}\n \\pgfplotsset{ compat=newest,\n scaled ticks=false,\n yticklabel style={\n \/pgf\/number format\/precision=3\n },\n }\n \\begin{axis}[ xlabel= {$\\beta f$},\n ylabel={$e_\\dis$},\n scatter\/classes={\n\t\t SA={blue},\n\t\t HA100={green},\n\t\t PA={red}\n\t\t },\n \t\t \/pgf\/number format\/precision =4\n \t\t]\n \\addplot[ scatter,only marks, mark size = 1.0pt,%\n scatter src=explicit symbolic\n ]\n table [meta =alg, x=bf, y=ed, col sep=space] {Plots\/bf_scatters_1600.txt};\n \\legend{$SA$,$HA$,$PA$}\n \\end{axis}\n \\end{tikzpicture}\n \\caption{Scatter plot of the dimensionless free energy per spin, $\\beta f$ and the disordered phase energy per spin $e_\\dis$ for each of $M=40$ runs for SA (blue), HA (green), and PA (red) at the transition temperature for system size $L=40$.}\n \\label{fig:bf_ed_scatter}\n\\end{figure}\n\n\\begin{figure}\n \\begin{tikzpicture}\n \\pgfplotsset{compat=newest,\n scaled ticks=false,\n yticklabel style={\n \/pgf\/number format\/.cd,\n fixed,\n fixed zerofill,\n \/tikz\/.cd\n },\n }\n \\begin{axis}[xlabel= {$\\beta f$}, ylabel={$\\A$},scatter\/classes={%\n \t\tSA={blue},%\n \t\tHA100={green},%\n \t\tPA={red}},%\n \t\t\/pgf\/number format\/precision = 4\n \t\t]\n \\addplot[scatter,only marks, mark size = 1.0pt,%\n scatter src=explicit symbolic]%\n table [meta=alg, x=bf, y=A, col sep=space]{Plots\/bf_scatters_1600.txt};\n \\legend{$SA$,$HA$,$PA$}\n \\end{axis}\n \\end{tikzpicture}\n \\caption{Scatter plot of the dimensionless free energy per spin, $\\beta f$ and disordered phase excess, $\\A$ for each of $M=40$ runs for SA (blue), HA (green), and PA (red) at the transition temperature for $L=40$. The exact value in thermodynamic limit, $X=0$, corresponds to peak ratio $\\rc=q$.}\n \\label{fig:bf_X_scatter}\n\\end{figure}\n\n\\begin{figure}\n \\hspace{-12pt}\n \\begin{tikzpicture}[yscale=.9, xscale = .9]\n \\begin{axis}[%\n grid=major, grid style = {dotted},\n width=3in,\n height=3in,\n scale only axis,\n xmin=-2,\n xmax=-0.2,\n every y tick label\/.append style={font=\\color{black}},\n ymin=0,\n ymax=3400,\n ytick ={ 0, 800, 1600, 2400, 3200},\n ylabel ={$\\tau$},\n xlabel ={$e$},\n yticklabels ={ 0, 800, 1600, 2400, 3200},\n xtick ={-.4,-.8,-1.2, -1.6, -2.0},\n xticklabels ={-.4,-.8,-1.2,-1.6,-2.0},\n legend pos= north west\n ]\n \\addplot [\n color=red,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=T, col sep=space]{Plots\/autocorrelation_30.txt};\n \\addlegendentry{$L=30$}\n \\addplot [\n color=blue,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=T, col sep=space]{Plots\/autocorrelation_40.txt};\n \\addlegendentry{$L=40$}\n \\addplot [\n color=green,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=T, col sep=space]{Plots\/autocorrelation_50.txt};\n \\addlegendentry{$L=50$}\n \\addplot [\n color=black,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=T, col sep=space]{Plots\/autocorrelation_60.txt};\n \\addlegendentry{$L=60$}\n \\end{axis}\n \\end{tikzpicture}\n \\caption{Integrated autocorrelation time $\\tau$ in units of Monte Carlo sweeps, as a function of energy ceiling per spin $e$ for system sizes $L=30$, 40, 50 and 60.}\n \\label{fig:autocorrelation_by_size}\n\\end{figure}\n\n\\begin{figure}\n \\hspace{-12pt}\n \\begin{tikzpicture}[yscale=.9, xscale = .9]\n \\begin{axis}[\n grid=major, grid style = {dotted},\n xmode=log,\n ymode=log,\n xlabel={cluster size},\n ylabel={$\\csh$},\n ]\n \\addplot [\n color=red,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=size, y=csh, col sep=space]{Plots\/cluster_size_histogram_0.txt};\n \\addlegendentry{$e=-1.56$}\n \n \\addplot [\n color=blue,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=size, y=csh, col sep=space]{Plots\/cluster_size_histogram_1.txt};\n \\addlegendentry{$e=-1.73$}\n \n \\addplot [\n color=green,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=size, y=csh, col sep=space]{Plots\/cluster_size_histogram_2.txt};\n \\addlegendentry{$e=-1.84$}\n \n \\addplot [\n color=red,\n dashed,\n line width=1.0pt,\n ] table[mark=none, dashed, x=size, y=fit, col sep=space]{Plots\/cluster_size_histogram_0.txt};\n \n \\addplot [\n color=blue,\n dashed,\n line width=1.0pt,\n ] table[mark=none, dashed, x=size, y=fit, col sep=space]{Plots\/cluster_size_histogram_1.txt};\n \n \\end{axis}\n \\end{tikzpicture}\n \\caption{Disordered cluster size histograms, $\\csh$ for ceiling energies per spin $e=-1.56$ (red), $-1.73$ (blue), and $-1.84$ (green) showing the behavior of $\\csh$ above, near, and below the evaporation transition energy $e_4$, respectively, for size system $L=40$. Dashed lines represent fitted power laws to the histograms for cluster sizes below a cutoff of 35.}\n \\label{fig:csh}\n\\end{figure}\n\n\\begin{figure}[h]\n \n \\hspace{-12pt}\n \\begin{tikzpicture}[yscale=.9, xscale = .9]\n \\begin{axis}[%\n grid=major, grid style = {dotted},\n width=3in,\n height=3in,\n scale only axis,\n \n xmin=-2,\n xmax=-0.2,\n every y tick label\/.append style={font=\\color{black}},\n ymin=0,\n ymax=2200,\n ytick ={ 0, 500, 1000, 1500, 2000},\n ylabel ={$\\tau\\ (\\textrm{sweeps})$},\n xlabel ={$e$},\n yticklabels ={ 0, 500, 1000, 1500, 2000},\n xtick ={-.4,-.8,-1.2, -1.6, -2.0},\n xticklabels ={-.4,-.8,-1.2,-1.6,-2.0},\n ]\n \\addplot [\n color=red,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=T, col sep=space]{Plots\/autocorrelation_time.tex};\n \\addlegendentry{$\\tau$}\n \\label{autocorrelation}\n \\end{axis}\n \\begin{axis}[%\n width=3in,\n height=3in,\n scale only axis,\n axis x line*=none,\n axis y line*=right,\n ylabel={$\\omega$},\n ylabel near ticks, yticklabel pos=right,\n every y tick label\/.append style={font=\\color{black}},\n ymin=0,\n ymax=2.2,\n xmin=-2.0,\n xmax=-0.2,\n ytick ={ 0, .5, 1.0, 1.5, 2.0},\n yticklabels ={ 0, 0.5, 1.0, 1.5, 2.0},\n xtick ={-.4,-.8,-1.2, -1.6, -2.0},\n xticklabels ={-.4,-.8,-1.2,-1.6,-2.0},\n ]\n \\addlegendimage{\/pgfplots\/refstyle=autocorrelation}\n \\addlegendentry{$\\tau$}\n \\addplot [\n color=blue,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=w, col sep=space]{Plots\/wrapping.tex};\n \\addlegendentry{$\\omega_w$}\n \\addplot [\n color=green,\n solid,\n line width=1.0pt,\n ] table[mark=none, x=e, y=c, col sep=space]{Plots\/condensation.tex};\n \\addlegendentry{$\\omega_c$}\n \\end{axis}\n \\end{tikzpicture}\n \\caption{The integrated autocorrelation time $\\tau$ (red, left axis) is plotted against ceiling energy per spin $e$ over the entire simulation energy range $-2 < e < 0$ for $L=40$. The wrapping fraction $\\omega_w$ (blue, right axis) is plotted from $-1.6 < e < .7 $ and the droplet fraction $\\omega_c$ (green, right axis) is plotted from $-1.85 < e < -1.5$. The first two peaks in $\\tau$, from right to left, coincide with jumps in $\\omega_w$ that signify the first and second wrapping transitions, respectively. The third peak in $\\tau$ coincides with a rapid drop of $\\omega_c$ signifying the evaporation transition. \n }\n \\label{fig:autocorrelation}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\nWe have introduced a class of equilibrium annealing algorithms that includes population annealing and an equilibrium version of simulated annealing. We have implemented algorithms in this class in the microcanonical ensemble and applied them to the 20-state Potts model, which displays a strong first-order transition. In agreement with previous work, we find that simulating a thermal first-order transition can be effectively carried out in the microcanonical ensemble yielding high precision results. In these applications of microcanonical annealing, the purpose of annealing is to collect data at all energies in the coexistence region in order to obtain canonical averages at the phase transition temperature using reweighting.\n\nWe compared the performance of equilibrium simulated annealing, population annealing and a hybrid algorithm that interpolates between the two. For the system sizes and number of sweeps considered here, equilibrium simulated annealing was found to be the most efficient algorithm followed by population annealing. The hybrid algorithm performed worst except for the largest size ($L=60$) where it slightly outperformed population annealing, though for this size neither algorithm equilibrated the system with the allotted number of Monte Carlo sweeps. \n\nWe conjecture that SA outperforms PA and HA because it takes advantage of the exponential convergence to equilibrium of Markov chain Monte Carlo in contrast to the $1\/R$ convergence to equilibrium of sequential Monte Carlo. Note that for $L=40$ the magnetization integrated autocorrelation time $\\tau$ is bounded by $\\tau \\leq 2000$ while the number of sweeps per annealing step in this region is $7.6 \\times 10^6$ so there are more than $10^3\\tau$ sweeps at every energy and, assuming the rate of convergence to equilibrium is approximately $\\tau$, systematic errors will be extremely small. By contrast, in PA the number of sweeps per annealing step carried out in the transition region is much less than $\\tau$ and equilibration is achieved primarily by resampling, which converges only as $1\/R$. The hybrid algorithm suffers from both an inadequate population size for effective resampling and too few Monte Carlo sweeps. \n\nA toy model illustrates the above intuition. Suppose that the distance from equilibrium, $\\Delta$ behaves as $\\Delta = (1\/R)e^{-t\/\\tau}$ where $R$ is the population size, $t$ the number of MCMC sweeps for each replica at each temperature and $\\tau$ is the autocorrelation time of the MCMC, which is here assumed to be constant. We allot each algorithm the same total number of sweeps per annealing step so that the scaled number sweeps, $c$, defined as $c=Rt\/\\tau$ is held fixed. Note that $c$ is the number of autocorrelation times for each annealing step in simulated annealing. In terms of $R$ and $c$ we seek to minimize $(1\/R)e^{-c\/R}$ for fixed $c$ subject to the constraints that the population size is at least one, $R \\geq 1$, and that at least one MCMC sweep is performed on each replica at each annealing step, $t \\geq 1$. The latter constraint is required to insure that some exploration and decorrelation is carried out at each annealing step. In terms of population size, this constraint can be written as $R \\leq c\\tau$. It is straightforward to show that there are two minima for $\\Delta$ and they occur at the two endpoints, SA ($R=1$), and PA ($R = c\\tau$). At the SA endpoint $\\Delta = e^{-c}$ while at the PA endpoint $\\Delta \\approx 1\/c\\tau$. In addition there is a maximum at $R=c$ where $\\Delta = (1\/ec)$. While this toy model cannot be expected to provide quantitative results, it is expected to yield qualitative comparisons between algorithms.\n\nThe first conclusion from the toy model is that if the autocorrelation time is sufficiently short that $t\/\\tau$ can be made large, then it is better to use SA.\nA second conclusion is that it does not pay to compromise between many MCMC sweeps and a large population. In our case $c$, at least in the difficult coexistence region, is approximately $10^3$ so that $\\Delta_{\\rm SA} \\ll \\Delta_{\\rm PA}$ and we expect that SA is the preferred algorithm. The hybrid algorithm with $R=10^2$ is not far from the worst choice, $R=c$ while PA is closer to the second optimum at $R=c \\tau$ so we expect that for our choices of parameters HA would perform worse that PA. \n\nWe can conclude from both the simulation results and the simple toy model that SA is the best of the microcanonical annealing methods for high precision studies of the transition in large-$q$ Potts models and, likely, other systems with thermal first-order transitions and simple symmetry breaking. \nOn the other hand, for systems that display a rough free energy landscape without obvious symmetries, we expect PA or HA to outperform SA. In each run of SA for the $q$-state Potts model only one of the $q$ low temperature phases is discovered because the barriers between phases are too high. This deficiency does not affect any of the observables we measured because of the symmetry between the $q$ low temperature phases. The situation is different in the case of multiple inequivalent phases or a glassy phase with a rough free energy landscape. In order to apply SA to a system with a free energy landscape with multiple inequivalent minima separated by high barriers it is necessary to combine many runs of SA using weighted averaging. Good statistics requires a large number of runs so that each important minimum is found in many runs. We conjecture that it is better to do a single run of HA or PA with the same annealing schedule as for SA and with $R=M$ replicas where $M$ is the number of runs used for SA. For HA or PA, the resampling step distributes the work assigned to each minimum according to the weight of that minimum so that the more important minima are more accurately sampled and minima that are irrelevant at the target low temperature are removed from the population. Evidence supporting this conjecture can be found in Ref.\\ \\cite{WaMaKa15}, where it was shown that PA with population size $R$ is far more likely to find ground states of Ising spin glasses than $M=R$ runs of SA with the same annealing schedule. \n\nThe microcanonical annealing algorithms discussed here may also be useful for sampling equilibrium systems with rough free energy landscapes and solving hard combinatorial optimization problems in situations where behavior similar to first-order transitions occurs. For example, spin glasses display temperature chaos \\cite{McBeKi82,BrMo87,WaMaKa15a}, which is the phenomena that typical states of the system changes discontinuously with temperature. It would be interesting to see whether microcanonical population annealing performs better than canonical population annealing for systems with temperature chaos.\n\n\\acknowledgements\nThis work was supported in part by the National Science Foundation (Grant No.~DMR-1507506). We thank Chris Amey, Gili Rosenberg, Martin Weigel, Roman Koteck{\\'{y}}, and Helmut Katzgraber for useful discussions. \n\n\n\\bibliographystyle{unsrt}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecent astrophysical observations of type Ia supernovae (SNIa) \\cite%\n{Perlmutter:1998np}, cosmic microwave background (CMB) \\cite{Komatsu:2010fb}, large scale structure (LSS) \\cite{Tegmark2004}, and the recent Planck measurements \\cite{Planck}, suggest that our\nuniverse is undergoing an era of accelerated expansion. Quantitative\nanalysis shows that there is a dark energy component (DE) with a negative pressure\ncomponent leading to the current accelerating expansion\nof the Universe. Since the fundamental origin and nature of such a dark energy remain\nenigmatic at present, various models of dark energy have been put forward, such as a\nsmall positive cosmological constant\\ \\cite{Peebles} and several kinds of\nscalar fields like quintessence \\cite{Ratra}, k-essence \\cite%\n{ARMENDARIZ-PICON2001}, phantom \\cite{Caldwell}, etc. \n{ Furthermore, the price of explaining the current cosmic acceleration by DE is the appearence of future singularities at late-time Universe. While some singularities like a big rip (BR) \\cite{Caldwell}, sudden singularities \\cite{Noj, Bar}, big freeze singularities \\cite{Bouh3} and big brake singularities \\cite{Chim1,Cata} could happen at a finite cosmic time, others abrupt which are events smoother like a little rip (LR) \\cite{Ruz, Noj2, Stef, Bouh1, Fram, Bre, Bouh2}, a little sibling of a BR \\cite{Yas1}, a little bang and a little sibling of big bang \\cite{Yas2} could happen at a infinite cosmic time.}\n\nCurrently, another model inspired by the holographic principle has been put\nforward to explain the current cosmic acceleration, which states that the number of degrees\nof freedom for a system within a finite region should be finite and bounded\nby the area of its boundary \\cite{'tHooft:1993gx, Susskind:1994vu}.\nIt is commonly believed that the holographic principle\n\\cite{'tHooft:1993gx, Susskind:1994vu, Bekenstein} is a fundamental principle\nof quantum gravity. Based on an effective quantum field theory, Cohen et\nal. \\cite{Cohen:1998zx} pointed out that, for a system with size L, which is not a black hole, the quantum vacuum energy of the system\nshould not exceed the mass of the same size black hole, i.e. $L^{3}\\rho\n_{\\Lambda }\\leq LM_{p}^{2}$, where $\\rho_{\\Lambda }$ is the vacuum energy\ndensity fixed by UV cutoff $\\Lambda $ and $M_{p}$ denotes the Plank mass.\nThe largest IR cutoff $L$ is chosen by saturating the\ninequality \\cite{Li:2004rb, Hsu:2004ri} so that we get the\nholographic energy density $\\rho_{H}=3c^{2}M_{p}^{2}\/L^{2}$, where $c$\nis a numerical factor. Later on and for convenience, we will instead use the parameter defined\nas $\\beta =c^{2}$. The holographic dark energy model is based on applying the previous ideas to the universe as a whole with the goal of explaining the current speed up of the universe. Then, the IR cutoff can be taken as a\ncosmic scale of the universe, like the Hubble horizon \\cite{Li:2004rb,Hsu:2004ri},\nparticle horizon, event horizon \\cite{Li:2004rb} or second order geometrical invariants \\cite{luongo1}. Another\nchoice for the IR cutoff L was suggested by Gao et al. \\cite{Gao} (see also \\cite{NojiriGRG2006}),\nin which the IR cutoff of the holographic dark energy (HDE) is taken\nto be the Ricci scalar curvature. Being an invariant quantity, the Ricci scalar curvature has many particularities such as its cosmological implications in describing the HDE \\cite{Yas2,Gao,NojiriGRG2006,OualiPRD85}, its space-time dependence and its advantages in avoiding the fine tuning and the causality problems \\cite{Li:2004rb}. For all these reasons we will choice the Ricci scalar as the holographic cutoff.\n\n\nAnother approach to explain the observed acceleration of the late universe is\nbased on alternative theory of general relativity like the one inspired by string theory such as the brane-world scenario. The induced gravity brane-world model proposed by Dvali, Gabadadze,\nand Porrati (DGP) is well known and studied \\cite{Dvali:2000hr}. It contains two branches \\cite{Deffayet:2000uy} the self accelerating branch which suffers from some problems and the normal branch. Even though the normal branch is healthy it cannot describe the current acceleration of the universe unless a dark energy component is invoked \\cite{Sahni,Zhang} or the gravitational action is modified \\cite{BouhmadiLopez:2010pp}. In the context of the DGP scenario, different models have been studied with various kind of sources for DE in Refs \\cite{jawad,dutta,Rav,far,she,gha,Shtanov1}. \\\\\n\nFurthermore, in the context of the dark sector, one of the main problem raised and without explanation in the framework of $\\Lambda$CDM cosmology is the cosmic coincidence problem, i.e. why dark energy density is of the same order of magnitude as cold dark matter energy density (CDM). An interacting mechanism between these two components could alleviate the cosmic coincidence problem as suggested by several authors \\cite{Bouhmadi-Lopez:2016dcs,Morais:2016bev,AmendolaPRD62, BoehmerPRD78, ChenPRD78, PavonPLB628, CampoPRD78, DuranPRD83, Rav}. These kind of interactions have been invoked in \\cite{AbdallaPRD95} in order to explain the possible departure from the $\\Lambda$CDM model as measured recently by the experiment of the BOSS \\cite{delubacAA574} for a value of the Hubble parameter at redshift $z=2.34$.\nIn Ref. \\cite{OualiPRD85}, we pointed out that the normal branch when filled with an holographic Ricci dark energy (HRDE) can face some DE singularities. This motivated us to improve the model with the aim to remove or smooth these singularities by introducing an interaction between the HDE density and the CDM sector. An interaction between DE and CDM can relieve the coincidence problem and at the same time may smooth some DE singularities. \\\\\n{Recently, interactions between DE and CDM in the holographic model have received a great interest \nby choosing the infrared cutoff as the Hubble scale \\cite{PavonPLB628, shey}, as the future event horizon \\cite{WangPLB624}, as the Granda and Oliveros scale \\cite{jawad}, as a Ricci scale \\cite{Yas2} and as a modified holographic DE \\cite{Chim2,Chim3}}\nWe show that this kind of interaction is also a promising way to avoid or to smooth the big rip and the little rip appearing in the non interacting models \\cite{Li:2004rb, Hsu:2004ri, OualiPRD85, BouhmadiPRD84}. \\\\\n\nThe unknown nature of DE and CDM makes difficult and imprecise the choice of the form of the interaction between them. However, the interaction is usually considered from a phenomenological point of view \\cite{WangPLB624,GuoPRD76,OlivaresPRD74,Wang:2016lxa}, from the outset given in \\cite{PavonPLB628, CampoPRD78} or from thermodynamical consideration \\cite{WangPLB662,PavonGRG41}. The conservation equations have dimensions of energy density divided by unit of time, therefore, the interaction between DE and CDM is expected to lead precisely to this kind of terms on the right hand side (rhs) of their respective continuity equations, i.e. functions of the energy densities of DE and CDM multiplied by a quantity with units inverse of time such as the Hubble scale as have been widely discussed in Refs \\cite{WangPLB624, GuoPRD76, WangPLB637, PavonPLB628, CampoPRD74, CampoPRD71, OlivaresPRD74}. The cosmological perturbations in this kind of models have been studied in \\cite{HePLB}. It was shown in that work that if the energy density of DE is phantom like, the curvature perturbations are always stable no matter if the coupling is proportional to DE density or to CDM energy density. However, it was shown that if dark energy is of a quintessence nature, the curvature perturbations are unstable unless the coupling of the interaction is proportional to DE density and the range of the coupling takes some specific values.\\\\\n\nMotivated by the study of Refs. \\cite{HePLB,AbdallaPRD95}, we consider as well this form of interaction i.e. $Q = \\lambda_mH\\rho_m$, $Q = \\lambda_HH\\rho_H$, or $Q = H(\\lambda_m\\rho_m+\\lambda_H\\rho_H)$ where $Q$ denotes the interaction between the energy densities $\\rho_m$ of CDM and\nthe HRDE component $\\rho_H$. The range of the coupling of the interaction, $\\lambda_m$ and $\\lambda_H$, are determined by observations \\cite{HePRD83,FengPLB665}. Considering that there is only energy transfer between DE and CDM, the energy transfer is from CDM to DE if $Q<0$ or from DE to CDM if $Q>0$ (see Eqs. (\\ref{EOSH}) and (\\ref{EOSm})). \\\\\n\nThe main aim of this paper is to show that an interacting holographic Ricci dark energy (IHRDE) with CDM can describe suitably the late-time acceleration of the universe, and at the same time improves the model without interaction by avoiding the big rip and\/or little rip happening in the model we studied in Ref. \\cite{OualiPRD85}.\\par\n\nThe outline of this paper is as follows. In Sec. II we briefly present an interacting CDM-HRDE model within a DGP brane-world model.\nWe assume that the Ricci scalar is the IR cutoff of the holographic energy density. In Sec. III, we study the modified Friedmann equation without a bulk\nGauss-Bonnet (GB) term by analyzing analytically the asymptotic\nbehavior of the brane and numerically the whole\nexpansion of the brane. An appropriate choice of the interaction \ncoupling $\\lambda_{H}$ avoids the big rip and the little rip from the normal\nbranch and hence the IHRDE gives a satisfactory and an alternative description of\nthe late time cosmic acceleration of the universe as compared with the HRDE without\nthe GB term in the bulk in the absence of interaction. Indeed the later\none modifies the big rip and little rip into a big freeze one\nwhile the former removes them definitively.\nIn Sec. IV, we consider the model\nwhere the bulk contains a GB term.\nIn this case the asymptotic behaviour of the IHRDE model depends on the sign of a discriminant ${\\mathcal{D}}$ which depends on the holographic parameter $\\beta $, the GB term, and the coupling $\\gamma$ (see\nEqs. (\\ref{factorD})-(\\ref{coeffF})). The IHRDE model succeed in removing the big rip and little\nrip from the future evolution of the brane. On this case, the brane will evolve\nasymptotically as a de Sitter universe for $\\gamma =\\frac{1}{2\\beta_{\\lim }}$,\nwhile for $\\gamma \\neq \\frac{1}{2\\beta_{\\lim }}$, the situation becomes more complicated and depends on\nthe values of the holographic parameter as well as on the interaction parameter. At this\nregard, as we will show that the coupling between HRDE and CDM plays a crucial role, even more important than the\nGB parameter, in removing the\nfuture singularities. Finally, in Sec. V, we conclude.\n\n\n\\section{\\ INTERACTING MODEL AND PARAMETER CONSTRAINTS}\n\nWe consider a DGP brane-world model, where the bulk contains a GB curvature\nterm, and the brane contains an induced gravity term on its action \\cite{Kofinas:2003rz,BouhmadiLopez:2008nf}.\nWe restrict our analysis to the normal branch. Assuming only an interaction between CDM and the\nholographic component on the brane, the conservation equations of the energy density read\n\n\\begin{eqnarray}\n\\overset{\\cdot }{\\rho }_{\\mathrm{H}}+3H(1+\\omega_{\\mathrm{H}})\\rho_{%\n\\mathrm{H}} &=&-Q \\label{EOSH} \\\\\n\\overset{\\cdot }{\\rho }_{\\mathrm{m}}+3H\\rho_{\\mathrm{m}} &=&Q. \\label{EOSm}\n\\end{eqnarray}\nwhere $H$ \\ is the Hubble parameter and $\\omega_{\\mathrm{H}}$ denotes the equation of state parameter of the holographic\ndark energy component.\n\nThe modified Friedmann equation in the normal branch of the DGP brane world universe containing an\nholographic Ricci dark energy, $\\rho_H$, and a CDM component, with energy density $\\rho_m$, can be written as\n\\cite{Kofinas:2003rz,BouhmadiLopez:2008nf,richard}\n\n\\begin{equation}\nH^{2}=\\frac{1}{3M_{p}^{2}}\\rho -\\frac{1 }{r_{c}}\\left( 1+\\frac{%\n8\\alpha }{3}H^{2}\\right) H, \\label{friedmann1}\n\\end{equation}%\nwhere $\\rho =\\rho_{\\mathrm{m}}+\\rho_{%\n\\mathrm{H}}$ is the total cosmic fluid energy density of the brane. The parameter $r_{c}$ is the cross-over scale which determines the\ntransition from a 4-dimensional (4D) to a 5-dimensional (5D) behaviour, and $\\alpha $ is the Gauss-Bonnet parameter.\n\nFurthermore, for a spatially flat Friedmann-Lema\\^{\\i}tre-Robertson-Walker universe, the Ricci scalar curvature is given by\n\n\\begin{equation}\n\\mathcal{R=}-6\\left( \\dot{H}+2H^{2}\\right),\n\\end{equation}%\nthe dot stands for the derivative with respect to the cosmic time of the\nbrane.\\\\\n\nAs already mentioned in the introduction, $\\rho_{\\mathrm{H}}$ is related to\nthe UV cutoff, while $L$ is related to the IR cutoff. Identifying $L^{-2}$\nwith $-\\mathcal{R}\/6,$ the energy density of the HRDE is given by \\cite{Gao,OualiPRD85}\n\\begin{equation}\n\\rho_{\\mathrm{H}}=3\\beta M_{P}^{2}\\left( \\frac{1}{2}\\frac{dH^{2}}{dx}%\n+2H^{2}\\right) , \\label{HRDE}\n\\end{equation}%\nwhere $x=\\ln (a\/a_{0})$. The quantities $a$, $a_0$ and $\\beta =c^{2}$ are respectively the scale factor, its\npresent value and the holographic dimensionless parameter\nwhich, as we will show, plays a significant role in determining the\nasymptotic behavior of the HRDE and therefore of the brane evolution. From now on, the subscript 0 stands for quantities evaluated at the present time.\n\nThe modified Friedmann equation\\ ({\\ref{friedmann1}) }can be further\nrewritten as:\n\n\\begin{equation}\nE^{2}=\\Omega_{\\mathrm{m}}+\\Omega_{\\mathrm{H}}-2\\sqrt{\\Omega_{%\n\\mathrm{r_c}}}(1+\\Omega_{\\mathrm{\\alpha }}E^{2})E, \\label{friedmann2}\n\\end{equation}%\nwhere $E(z)=H\/H_{0}$, $z$ is the redshift and\n\\begin{eqnarray}\n\\Omega_{\\mathrm{m}} &=&\\frac{\\rho_{\\mathrm{m}}}{3M_{p}^{2}H_{0}^{2}}%\n,\\,\\,\\,\\,\\Omega_{\\mathrm{r_c}}=\\frac{1}{4r_{\\mathrm{c}}^{2}H_{0}^{2}},\\text{%\n\\thinspace\\ } \\\\\n\\,\\Omega_{\\mathrm{\\alpha }} &=&\\frac{8}{3}\\alpha H_{0}^{2}, \\\\\n\\text{\\ }\\Omega_{\\mathrm{H}} &=&\\beta (\\dfrac{1}{2}\\dfrac{dE^{2}}{dx}%\n+2E^{2}). \\label{OmegaH}\n\\end{eqnarray}\n\n The cosmological parameters of the model are constrained by\nevaluating the Friedmann equation ({\\ref{friedmann2})} at present\n\n\\begin{equation}\n1=\\Omega_{\\mathrm{m}_{0}}+\\Omega_{\\mathrm{H}_{0}}-2\\sqrt{\\Omega\n_{r_c}}(1+\\Omega_{\\mathrm{\\alpha }}), \\label{present Friedmann}\n\\end{equation}\ni.e. $E(x=0)=1$. By combining this equation and Eq. (\\ref{OmegaH}) we obtain:\n\n\\begin{equation}\n\\frac{dE}{dx}\\Big|_{x=0}=-2+\\frac{\\Omega_{\\mathrm{H}_{0}}}{\\beta }\\text{,}%\n\\label{E'0}\n\\end{equation}%\nOn the other hand, given that the universe is currently accelerating the present value of the deceleration parameter, $q=-(1+d\\ln E\/dx)$, which reads:\n\\begin{equation}\nq_{0}=1-\\frac{\\Omega_{\\mathrm{H}_{0}}}{\\beta }. \\label{q0}\n\\end{equation}%\nmust be negative, therefore\n\n\\begin{equation}\n0<\\beta =\\frac{1-\\Omega_{\\mathrm{m}_{0}}+2\\sqrt{\\Omega_{rc}}%\n(1+\\Omega_{\\mathrm{\\alpha }})}{1-q_{0}}<\\Omega_{\\mathrm{H}_{0}}.\n\\label{beta}\n\\end{equation}\n\nAs we mentioned in the introduction, the following\nanalysis will be devoted to the normal branch because it requires dark energy and because it is free from the\ntheoretical problems plugging the self-accelerating branch. By using the constraint ({\\ref{present Friedmann})}, it can be shown that \nthe holographic parameter $\\beta$ verifies \n\\begin{equation}\n \\frac{%\n1-\\Omega_{\\mathrm{m}_{0}}}{1-q_{0}}<\\frac{1-\\Omega_{\\mathrm{m}_{0}}+2\\sqrt{\\Omega_{rc}}}{1-q_{0}}<\\beta.\n\\label{inequalities}\n\\end{equation}%\nTaking into account the latest Planck data \\cite{Planck}: $\\Omega_{\\mathrm{m}_{0}}\\sim{0.315}$ and $q_{0}\\sim -0.558$, therefore the\nratio $\\beta_{\\textrm{lim}}\\doteq\\frac{1-\\Omega_{\\mathrm{m}_{0}}}{1-q_{0}}$ is\nof the order $0.44$, which means that the normal branch is characterized by $0.44<\\beta$.\n\n From equations {(\\ref{friedmann2}) and (\\ref{OmegaH})} the\nvariation of the dimensionless Hubble rate $E$ with respect to $x$ is\n\n\\begin{equation}\n\\dfrac{dE^2}{dx}=-\\frac{2\\Omega_{m}+2(2\\beta -1)E^2-4\\sqrt{\\Omega_{r_c}}(1+\\Omega_{\\alpha\n}E^{2})E}{\\beta} \\label{variation of E}\n\\end{equation}\n\nOn the other hand, the energy conservation equation {(\\ref{EOSm})} gives\n\n\\begin{equation}\n\\dfrac{d\\Omega_{m}}{dx}+3\\Omega_{m}=\\lambda_{H}\\Omega_{H}+\\lambda\n_{m}\\Omega_{m}., \\label{EOSm1}\n\\end{equation}\n\nThen, using Eq. (\\ref{OmegaH}) we obtain\n\n\\begin{equation}\n\\dfrac{d\\Omega_{m}}{dx}=\\lambda_{H}\\left(\\dfrac{\\beta }{2}\\dfrac{dE^{2}}{dx}%\n+2\\beta E^{2}\\right)+\\left(\\lambda_{m}-3\\right)\\Omega_{m}, \\label{EOSm2}\n\\end{equation}\n\nand with Eq. (\\ref{variation of E}) \\ we get\n\n\\begin{eqnarray}\n\\dfrac{d\\Omega_{m}}{dx} &=&\\Big[2\\lambda_{H}\\beta -\\left( 2\\beta\n-1\\right) (\\lambda_{m}-3)\\Big] E^{2} \\notag \\\\\n&+&\\Big[ \\lambda_{H}\\beta -\\beta (\\lambda_{m}-3)\\Big] E\\dfrac{dE}{dx}\n\\label{variation of Omega m} \\\\\n&+2&\\sqrt{\\Omega_{r_c}}(\\lambda_{m}-3)(1+\\Omega_{\\alpha }E^{2})E,\n\\notag\n\\end{eqnarray}\n\nThe derivative of equation (\\ref{variation of E}) gives\n\n\\begin{eqnarray}\n\\dfrac{d^{2}E^{2}}{dx^{2}}&=&-\\frac{2}{\\beta }\\dfrac{d\\Omega_{m}}{dx}-\\frac{%\n2\\left( 2\\beta -1\\right) }{\\beta }\\dfrac{dE^{2}}{dx} \\notag \\\\\n&+&\\frac{4\\sqrt{\\Omega_{r_c}}}{\\beta }(1+3\\Omega_{\\alpha }E^{2})%\n\\dfrac{dE}{dx} \\label{Second variation of E}\n\\end{eqnarray}\n\nFinally, by using Eq. (\\ref{variation of Omega m}), we have\n\n\\begin{eqnarray}\n\\dfrac{d^{2}E^{2}}{dx^{2}} &=&\\Big[-\\lambda_{H}+(\\lambda_{m}-3)-2\\frac{\\left(\n2\\beta -1\\right) }{\\beta }\\Big]\\dfrac{dE^{2}}{dx} \\notag \\\\\n&+2&\\Big[-2\\lambda_{H}+\\frac{\\left( 2\\beta -1\\right) (\\lambda_{m}-3)}{\\beta }%\n\\Big]E^{2} \\notag \\\\\n&+&\\frac{4\\sqrt{\\Omega_{r_c}}}{\\beta }(1+3\\Omega_{\\alpha }E^{2})%\n\\dfrac{dE}{dx} \\label{variation double de E} \\\\\n&-&\\frac{4\\sqrt{\\Omega_{r_c}}(\\lambda_{m}-3)}{\\beta }(1+\\Omega\n_{\\alpha }E^{2})E \\notag\n\\end{eqnarray}\n\n\nIn the absence of interaction, i.e. for $\\lambda_{H}=\\lambda_{m}=0$, the\nmodel coincides with that of Ref. \\cite{OualiPRD85}, in which we have\nshown that for $\\Omega_{\\alpha }=0$, the HRDE will have a phantom-like behaviour until it reaches a big rip\nsingularity for $\\beta <1\/2$ and a little rip for $\\beta =1\/2$. In our\npresent model, in which the interaction between CDM and HRDE is included, we can show\nnumerically that the little rip for $\\beta =1\/2$, and the\nbig rip for $\\beta_{\\lim }<\\beta <1\/2$ can be avoided which in the previous work \\cite{OualiPRD85}\nbecomes a big freeze by including a GB term in the bulk action. The question that\nnow arises is for which values of $\\lambda_{H}$, $\\lambda_{m}$ and $\\beta$ these singularities\nappearing in that model can be avoided? \n{ To answer this question, we notice that the differential equation ({\\ref{variation double de E})} is not linear so it is not easy to solve analytically. However, our numerical analysis of Eq. ({\\ref{variation double de E})} shows that its asymptotic behaviour is the same as the one of Eqs. ({\\ref{variation of Omega m})} with ${d\\Omega_m}\/{dx}=0$ i.e. in the far future $\\Omega_m$ as well as ${d\\Omega_m}\/{dx}$ can be neglected. This simplify considerably our task with respect of the above singularities.}\n\\section{ Model without a Gauss-Bonnet term}\n\nIn order to solve equation ({\\ref{variation double de E})}, we\nconsider first the model without a bulk Gauss-Bonnet term i.e. $\\Omega_{\\alpha }=0$, therefore\n\\begin{eqnarray} \\label{variation of E (dgp)}\n\\dfrac{d^{2}E^{2}}{dx^{2}} &=&\\Big[-\\lambda_{H}+(\\lambda_{m}-3)-2\\frac{\\left(\n2\\beta -1\\right) }{\\beta }\\Big]\\dfrac{dE^{2}}{dx} \\notag\\\\\n&+&2\\Big[-2\\lambda_{H}+\\frac{\\left( 2\\beta -1\\right) (\\lambda_{m}-3)}{\\beta }%\n\\Big]E^{2} \\notag \\\\\n&+&\\frac{4\\sqrt{\\Omega_{r_c}}}{\\beta }\\dfrac{dE}{dx}-\\frac{%\n4\\sqrt{\\Omega_{r_c}}(\\lambda_{m}-3)}{\\beta }E\n\\end{eqnarray}\nand we can write the equation ({\\ref{variation of Omega m})} as\n\n\\begin{eqnarray}\n&&\\left[ \\lambda_{H}\\beta -\\beta (\\lambda_{m}-3)\\right] \\dfrac{dE}{dx}\n\\label{diff eq} \\\\\n&=&-\\left[ 2\\lambda_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda_{m}-3)%\n\\right] E-2\\sqrt{\\Omega_{rc}}(\\lambda_{m}-3) \\notag\n\\end{eqnarray}\n\nIn the following, we will discus the solution to the above equation for the\nmodel $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$, $Q=\\lambda_{\\mathrm{H}%\n}H\\rho_{\\mathrm{H}}$, and $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}%\n}+\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}.$\n\n\\subsection{The model $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$}\n\nThis model corresponds to $\\lambda_{H}=0$ and can be split in two cases.\nThe case $\\lambda_{m}=3$ which implies, from Eq. {(\\ref{variation of Omega m})}, that $\\Omega_{m}$ is always constant. This case is not physically\nacceptable as DM behaves as a cosmological constant. And the case $\\lambda_{m}\\neq 3$ where the holographic parameter plays a crucial\nrole in determining the asymptotic behaviour of the brane:\\\\\n\\subsubsection{\\ Asymptotic behavior $\\beta =1\/2$}\nThere is a unique solution corresponding to little rip solution supported by the normal branch\n\\begin{equation}\nE=4\\sqrt{\\Omega_{rc}}x+C_{1},\n\\end{equation}\nwhere $C_{1}$ is a constant of integration. \\\\\n\n\\subsubsection{ Asymptotic behavior $\\beta \\neq 1\/2$:} \n\nEquation (\\ref{diff eq}) gives the solution%\n\\begin{eqnarray}\n&&\\left\\vert \\left( 2\\beta -1\\right) E-2\\sqrt{\\Omega_{r_c}}%\n\\right\\vert \\\\\n&=&\\left\\vert \\left( 2\\beta -1\\right) E_{1}-2\\sqrt{\\Omega_{r_c}}%\n\\right\\vert \\exp [-\\frac{\\left( 2\\beta -1\\right) }{\\beta }(x-x_{1})], \\notag\n\\end{eqnarray}\nwhere $E_1$ and $x_1$ are integration constants.\n\\begin{itemize}\n\\item For $\\beta >1\/2$, the dimensionless Hubble rate reaches a constant value and the brane is asymptotically de Sitter\n\\begin{equation}\nE_{\\infty }=\\frac{2}{2\\beta -1}\\sqrt{\\Omega_{rc}}.\n\\label{Einfinit m}\n\\end{equation}\n\nWe notice that this\nasymptotic de Sitter solution is possible only for $\\beta \\neq 1\/2$.\nOn the other hand, we note that equation ({\\ref{EOSm1}) with }$%\n\\lambda_{H}=0$, gives the solution $\\Omega_{m}=\\Omega_{\\mathrm{m}%\n_{0}}e^{(\\lambda_{m}-3)x},$ and by substituting it in Eq. ({\\ref{friedmann2}%\n), }we conclude that with the finite value of the dimensionless Hubble\nparameter $E_{\\infty }$ of Eq. ({\\ref{Einfinit m}),} $\\lambda_{m}$ must be\nless or equal to $3$ to ensure that$\\ \\Omega_{\\mathrm{H}}$ (and therefore $%\nE $) converges asymptotically in the far future to a finite value. So in this\ncase the energy density of CDM is practically zero at the far future, and the\nuniverse converges asymptotically to a universe filled exclusively with an\nHRDE component.\n\n\\item For $\\beta_{\\textrm{lim}}<\\beta <1\/2$, the dimensionless Hubble rate blows up in the\nfar future, and it follows a superaccelerated expansion until it hits a big\nrip singularity.\\\\\n\nTherefore, the IHRDE model with $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$\nhas the same asymptotic behaviour as the model analized in Ref. \\cite{OualiPRD85}.\nSo we conclude that this model does not succeed in removing the big rip and the little\nrip singularities happening in the non-interacting model \\cite{OualiPRD85}.\n\\end{itemize}\n\n\\subsection{The model $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$}\nThis model corresponds to $\\lambda_{{m}}=0$. The holographic $\\beta$ parameter determines the\nasymptotic behaviour of the brane as follow.\\\\\n\\subsubsection{Asymptotic behavior\n$\\beta =\\beta_{\\mathrm{LR}}\\doteq\\frac{3}{2(\\lambda_{H}+3)}$} \n\nIn this case Eq. (\\ref{diff eq}) gives a little rip solution:\n\n\\begin{equation}\nE_{\\mathrm{LR}}=4 \\sqrt{\\Omega_{rc}}x+C_{2}, \\label{LR solution}\n\\end{equation}\n\nwhere $C_{2\\text{ }}$is an integration constant.\\\\\n\n\\subsubsection{Asymptotic behavior $\\beta \\neq \\beta_{\\mathrm{LR}}$} \n\nThe solution of Eq. (\\ref{diff eq}) is given by:\n\n\\begin{eqnarray}\n&&\\left\\vert \\left[ 2\\lambda_{H}\\beta +3\\left( 2\\beta -1\\right) \\right]\nE-6\\sqrt{\\Omega_{rc}}\\right\\vert \\\\\n&=&\\left\\vert \\left[ 2\\lambda_{H}\\beta +3\\left( 2\\beta -1\\right) \\right]\nE_{1}-6\\sqrt{\\Omega_{rc}}\\right\\vert \\notag \\\\\n&&\\exp [-\\frac{\\left[ 2\\lambda_{H}\\beta +3\\left( 2\\beta -1\\right) \\right] }{%\n\\beta (\\lambda_{H}+3)}(x-x_{1})], \\notag\n\\end{eqnarray}\nwhere $E_1$ and $x_1$ are integration constants.\\\\\n\nFor clarity we divide our analysis in two cases:\n\\begin{enumerate}\n\\item { $\\beta <\\beta_{\\mathrm{LR}}$.}\\par\n\nThe dimensionless Hubble rate blows up in the far future, and it follows a\nsuper accelerated expansion until it hits a big rip singularity.\n\n\\item {$\\beta >\\beta_{\\mathrm{LR}}$.}\\par\n\nThe brane is asymptotically de Sitter, i.e. the Hubble rate reaches a constant value $E_{\\infty }$,\n\\begin{equation}\nE_{\\infty }=\\frac{6 \\sqrt{\\Omega_{rc}}}{2\\beta \\lambda_{H}+3\\left( 2\\beta\n-1\\right) }. \\label{Einfinit h}\n\\end{equation}%\n $E_{\\infty }$ must be positive. The condition $E_{\\infty }>0$, is directly related to the choice of the parameters of the model as can be seen in\nEq. ({\\ref{Einfinit h})}. For $\\beta =1\/2$, the solution reduces to $E_{\\infty }=%\n6\\sqrt{\\Omega_{rc}}\/\\lambda_{H}$ which is finite and no little rip is reached\nfor a finite $\\lambda_{H}$, unless when $\\lambda_{H}\\longrightarrow 0$, where $E_{\\infty }$ blows up and one approaches the model\nwhere there is no interaction \\cite{OualiPRD85}, for which the little rip is\ninevitable for $\\beta =1\/2$. For $\\beta >1\/2$ the asymptotical de Sitter\nsolution $E_{\\infty }$ remains even if $\\lambda_{H}=0.$\n\\end{enumerate}\n\nWe conclude that the interacting model acts only on the interval\n$\\beta_{\\textrm{lim}}<\\beta \\leq 1\/2$ of the normal branch, and by choosing $\\beta_{%\n\\mathrm{LR}}=\\frac{3}{2(\\lambda_{H}+3)}<\\beta_{\\textrm{lim}}$ one can avoid the\nbig rip singularity for $\\beta_{\\textrm{lim}}<\\beta <1\/2$ and the little rip for $\\beta =1\/2$\npresents in the model without interaction \\cite{OualiPRD85}.\nWe notice that the inclusion of interaction between CDM and\nHRDE gives a satisfactory results as compared with the\ninclusion of the GB effect in the HRDE model studied in Ref. \\cite{OualiPRD85}.\nIndeed, the inclusion of the GB term does not avoid the singularity but it\nalters it to a big freeze singularity while the interacting model smooth the singularities by\nan appropriate choose of the limiting value of the holographic parameter.\n\n\\subsection{The model $Q=\\lambda_{\\mathrm{H}}H%\n\\rho_{\\mathrm{H}}+\\lambda_{\\mathrm{m}}H\\rho_{%\n\\mathrm{m}}$}\n\n\\subsubsection{$\\lambda_{m}=3$}\n\nIn this case Eq. (\\ref{diff eq}) gives the solution of the dimensionless Hubble rate as\n\\begin{equation}\n E = E_{1} \\exp [-2(x-x_{1})],\n\\end{equation}\nwhere $E_1$ and $x_1$ are integration constants and it is reduced to a Minkowski one at far future.\n\n\\subsubsection{$\\lambda_{H}=(\\lambda_{m}-3)$ {and} $\\lambda_{m}\\neq 3$}\n\nIn this case the brane has a negative constant dimensionless Hubble rate and it is not physical at late-time\n\n\\begin{equation}\nE=-2\\sqrt{\\Omega_{rc}},\n\\end{equation}\n\n\\subsubsection{$\\lambda_{H}\\neq(\\lambda_{m}-3)$ {and} $\\lambda_{m}\\neq 3$}\n\n\\paragraph{Asymptotic behavior $\\beta =\\beta_{\\text{LR}%\n}\\doteq\\frac{3-\\lambda_{m}}{2\\lambda_{H}-2\\lambda_{m}+6%\n}$} \n\nWe notice that for $\\lambda_{m}\\neq 3$ and $\\lambda_{H}\\neq(\\lambda_{m}- 3)$ the\nsolution leads to a little rip when $\\beta =\\beta_{\\text{LR}}$ and the Hubble parameter reads\n\n\\begin{equation}\nE=4\\sqrt{\\Omega_{rc}}x+C_{3},\n\\end{equation}\n\nwhere $C_{3}$ is a constant of integration.\\\\\n\n\\paragraph{ Asymptotic behavior $\\beta \\neq\\beta_{\\text{LR}}$} \n\nThe solution of Eq. (\\ref{diff eq}) is given by:\n\n\\begin{eqnarray}\n&&\\left\\vert \\left[ 2\\lambda_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda\n_{m}-3)\\right] E+2\\sqrt{\\Omega_{rc}}(\\lambda_{m}-3)\\right\\vert \\notag\\\\\n&=&\\left\\vert \\left[ 2\\lambda_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda\n_{m}-3)\\right] E_{1}+2\\sqrt{\\Omega_{rc}}(\\lambda_{m}-3)\\right\\vert\n\\notag \\\\\n&&\\exp [-\\frac{\\left[ 2\\lambda_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda\n_{m}-3)\\right] }{\\lambda_{H}\\beta -\\beta (\\lambda_{m}-3)}(x-x_{1})]\n\\end{eqnarray}\n\\begin{enumerate}\n\n\\item If {$\\beta >\\beta_{\\text{LR}}$.}\n\nThe asymptotic solution corresponds to a de Sitter brane whose Hubble parameter reads\n\\begin{equation}\nE_{\\infty }=\\frac{-2\\sqrt{\\Omega_{rc}}(\\lambda_{m}-3)}{2\\lambda\n_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda_{m}-3)}. \\label{Einfinit2}\n\\end{equation}\n\nThis asymptotic de Sitter solution gives a finite asymptotic value of $%\n\\Omega_{\\mathrm{H}}$ (see Eq. (\\ref{OmegaH})).\\\\\n\nNotice that equation ({\\ref{EOSm1}) can be written }in the limit\nwhere\\ $\\Omega_{\\mathrm{H}}$\\ converges asymptotically to a finite\nvalue\\ $\\Omega_{\\mathrm{H}_{\\infty }}$ as $(3-\\lambda_{\\mathrm{m}})\\Omega\n_{\\mathrm{m\\infty }}=\\lambda_{\\mathrm{H}}\\Omega_{\\mathrm{H}_{\\infty }},${\\\nso }one can conclude that $\\lambda_{m}$ must be always less than $3$ in\norder to have $\\Omega_{\\mathrm{m\\infty }}>0$.\\\\\n\nOn the other hand, For $\\beta =1\/2$ the solution (\\ref{Einfinit2}) is reduced to\n$E_{\\infty }=\\frac{2\\sqrt{\\Omega_{rc}}(3-\\lambda_{m})}{\\lambda_{H}}$ which is\nfinite and the little rip is avoided for a finite $\\lambda_{H}$.\nWhile for $\\lambda_{H}\\longrightarrow 0$, $E_{\\infty }$ blows\nup. In this case one approaches the model $Q=\\lambda_{\\mathrm{m}}H\\rho_{%\n\\mathrm{m}}$ which coincides with the non interaction case \\cite{OualiPRD85}.\nFor $\\beta >1\/2$, the asymptotic de Sitter solution $E_{\\infty }$ is still\npresent even for $\\lambda_{H}=0$ and $\\lambda_{m}=0$. For $\\beta_{\\lim\n}<\\beta <0.5$, $E_{\\infty }$ is positive by choosing $\\lambda_{H}>\\frac{%\n1-2\\beta }{2\\beta }(3-\\lambda_{m})$. Here again by an appropriate choose of\n$\\lambda_{H}$, one can avoid the big rip singularity.\n\n\\item {$\\beta <\\beta_{\\text{LR}}$.}\n The dimensionless Hubble rate blows up in the future and it follows a super accelerating\n expansion until it reaches a big rip singularity.\\\\\n\\end{enumerate}\n\n\nAs in the previous subsection, we notice that the interaction between CDM and\nan holographic Ricci dark energy density gives\nsatisfactory results as compared with the inclusion of a Gauss Bonnet term in the bulk \\cite{OualiPRD85}.\nIndeed, the inclusion of a (GB) term does not avoid the singularity but it\nmodifies it to a big freeze singularity while the interacting model does.\nIt is worth noticing, from Eqs. (\\ref{diff eq}) and (\\ref{Einfinit2}), that the results of the model $Q=\\lambda_{%\n\\mathrm{H}}H\\rho_{\\mathrm{H}}+\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$ are\nsimilar to those with $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$ by making the\ntransformation $\\lambda_{H}$ $\\longrightarrow \\frac{3\\lambda_{H}}{%\n3-\\lambda_{m}}$ in the model $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$.\n\n\\subsection{ Numerical analysis}\n\nThe analytical solutions of equation ({\\ref{variation of E (dgp)}) are not at all\nobvious so a numerical analysis is required.}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG1.pdf}\n\\end{center}\n\\caption{{\\ Plot of the dimensionless Hubble rate $E$ against $x$. The\ncosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558 $, $\\lambda_{m}=0.1$ and $\\lambda_{H}=0$. The\nplot has been done for several values of the holographic parameter that are\nstated at the bottom of the figure.}}\n\\label{Em}\n\\end{figure}\nIn order to solve numerically Eq. ({\\ref{variation of E (dgp)})},\nwe choose $%\n\\Omega_{m0}=0.315$ and $q_{0}=-0.558$, as it is given by the latest Planck data\n\\cite{Planck} and assuming that our model is pretty much similar to a $%\n\\Lambda $CDM scenario at the present time. For a given value of $\\beta $,\nthe dimensionless energy density $\\Omega_{H_{0}}$ is fixed through Eq. ({%\n\\ref{q0})}, while the crossover scale parameter $\\Omega_{r_c}$ is fixed by\nthe constraint Eq. ({\\ref{present Friedmann})}.\n\nIn Fig. \\ref{Em}, we show the solutions corresponding to the model $Q=\\lambda\n_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$ for different values of $\\beta $. For $%\n{\\beta =0.45}$ and $\\beta =0.47$ the brane expands exponentially with respect\nto $x$ in the future until it reaches a big rip singularity, while for $\\beta =0.6$ and $%\n\\beta =0.8$ the brane expands as an inverse of the exponential of $x$ and remains\nasymptotically de Sitter in the future. Finally for $\\beta =0.5$\nthe brane has an asymptotic solution of the form $E=Ax+B$\nwhich corresponds to a little rip solution. All of these numerical\nsolutions are in agreement with the analytical analysis as it should be.\nIn addition, the model $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$ has the same\nasymptotic result as in Ref. \\cite{OualiPRD85}.\n\nFigs. \\ref{Eh} and \\ref{E} correspond to the model $Q=\\lambda_{\\mathrm{H%\n}}H\\rho_{\\mathrm{H}}$. In Fig. \\ref{Eh} we show the numerical\nsolutions of Eq. (\\ref{variation of E (dgp)}) for different values of $\\beta$.\nFor all these solutions the condition $\\beta >\\frac{3}{2(\\lambda_{H}+3)}$ is satisfied,\\ the brane\nis asymptotically de Sitter in the far future and the dimensionless Hubble\nrate $E$ approaches the value $E_{\\infty }=\\frac{6\\sqrt{\\Omega_{rc}}}{%\n2\\beta \\lambda_{H}+3\\left( 2\\beta -1\\right) }$ which is in agreement with\nthe analytical analysis (see Eq. (\\ref{Einfinit h})).\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG2.pdf}\n\\end{center}\n\\caption{{Plot of the dimensionless Hubble rate $E$ against $x$ for the\nnormal branch. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\lambda_{m}=0$ and $\\lambda %\n_{H}=0.4$. The plot has been done for several values of the holographic\nparameter that are stated at the bottom of the figure.}}\n\\label{Eh}\n\\end{figure}\nIt is worth noticing the avoidance of the big rip singularity for\n$\\beta_{\\textrm{lim}}<\\beta <1\/2$ and the little rip singularity for $\\beta =1\/2$ by\nassuming the interaction between CDM and HRDE component, while in the model \\cite{OualiPRD85} the big rip\nand the little rip singularities are obtained respectively for $\\beta <1\/2$ and $\\beta =1\/2$.\nFig. \\ref{E} illustrates the numerical solutions of Eq. (\\ref{variation\nof E (dgp)}) for $\\beta =0.45$ and for different values of $%\n\\lambda_{\\mathrm{H}}$. For \\ $\\lambda_{\\mathrm{H}}=1\/3$, $\\lambda_{\\mathrm{H}}>\\frac{1}{3}$\nand $\\lambda_{\\mathrm{H}}<\\frac{1}{3}$, Fig. \\ref{E} shows respectively the little rip solution, the de Sitter behaviour\nand the solution which hits a big rip singularity. For the model $Q=\\lambda_HH\\rho_H+\\lambda_m\\rho_m$, the\nabove discussions are translated to the sets of the couple $(\\lambda_m,\\lambda_H)$. Indeed, by translating the constraints\n on $\\lambda_H$ to $\\frac{3\\lambda_H}{3-\\lambda_m}$ and requiring that the couple $(\\lambda_m,\\lambda_H)$\n verifies these constraints the same conclusions are obtained.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG3.pdf}\n\\end{center}\n\\caption{{Plot of the dimensionless Hubble rate $E$ against $x$ for the\nbranch $\\beta =0.45$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558 $. The plot has been done for $%\n\\lambda_{m}=0$ and for several values of the parameter $\\lambda %\n_{H} $ that are stated at the bottom of the figure. We can see a big rip behaviour for $\\lambda_H=0.3$, little rip\nbehaviour for $\\lambda_H=1\/3$ and de Sitter behaviour for $\\lambda_H\\geq 0.35$.}}\n\\label{E}\n\\end{figure}\n\nIn order to complete our numerical study {and by imposing that the brane is currently accelerating,} we analyze the equation of state $\\omega_{H}=\\rm{p}_{H}\/\\rho_{H}$, $\\omega_{\\text{eff}}$\nand the deceleration parameter $q$ { where $\\rho_H$, $p_H$ and $\\omega_{\\text{eff}}$ are the HRDE density, its pressure and its effective equation of state associated to the effective energy density (please cf. Eq. (3.16)) respectively. }\n\nFrom Eq. (\\ref{EOSH}) and $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$, we\nobtain\n\\begin{equation}\n\\omega_{H}=-1-\\lambda_{H}-\\frac{1}{3}\\frac{d\\ln (\\Omega_{H})}{dx},\n\\end{equation}\nwhere $\\Omega_{H}$ is defined in Eq. (\\ref{OmegaH}). In terms of the\ndimensionless Hubble rate E and its derivatives with respect to $x$, $\\omega\n_{H}$ can be rewritten as\n\n\\begin{equation}\n\\omega_{\\mathrm{H}}=-\\lambda_{\\mathrm{H}}-1-\\dfrac{\\left( \\dfrac{dE}{dx}%\n\\right) ^{2}+E\\dfrac{d^{2}E}{dx^{2}}+4E\\dfrac{dE}{dx}}{3E\\dfrac{dE}{dx}%\n+6E^{2}}. \\label{eq wH}\n\\end{equation}\nThe effective equation of state $\\omega_{\\text{eff}}$ can be obtained by rewriting the Friedmann equation (\\ref{friedmann2}) as the standard Friedmann equation\n\\begin{equation}\\label{eff_friedmann}\n E^2=\\frac{1}{3M_p^2H_0^2}(\\rho_m+\\rho_{\\text{eff}}),\n\\end{equation}\nwhere $\\rho_{\\text{eff}}$ is the effective energy density that can be defined through Eq. (\\ref{HRDE}) as\n\\begin{equation}\\label{}\n \\rho_{\\text{eff}}= 3M_p^2H_0^2\\Big[\\frac{1}{2}\\beta\\frac{dE^2}{dx}+2\\beta E^2 -2\\sqrt{\\Omega_{\\mathrm{r_c}}}E\\Big].\n\\end{equation}\nFurthermore, from the conservation equation\n\\begin{equation}\n \\dot{\\rho}_{\\text{eff}}+3H(1+\\omega_{\\text{eff}})\\rho_{\\text{eff}}=0, \\label{rhoeff1}\n\\end{equation}\nwe obtain the effective equation of state\n\\begin{equation}\\label{}\n 1+\\omega_{\\text{eff}}=-\\frac{1}{3\\rho_{\\text{eff}}}\\frac{d\\rho_{\\text{eff}}}{dx}.\\label{rhoeff2}\n\\end{equation}\n\n\nFig. \\ref{w} shows some examples of the behavior of the equation of\nstate for the current cosmological values, for $\\lambda_{{H}}=0.1$ and for\ndifferent values of $\\beta $. As it can be clearly noticed from Eq. (\\ref{eq\nwH}) and illustrated in Fig. \\ref{w}, $\\omega_{H}$ approaches $-\\lambda_{%\n\\mathrm{H}}-1$ at very late-time for $\\beta>\\frac{3}{2(\\lambda_H+3)}$ corresponding\nto a de Sitter behaviour of the brane as it is also confirmed by the effective equation of state $\\omega_{\\text{eff}}$\nplotted in Fig. \\ref{weff} \n(for $\\lambda_{\\mathrm{H}}=0.1$ and $\\beta>0.48$, $\\omega_{H}$ approaches $-1.1$ and $\\omega_{\\text{eff}}$ is less than $-1$).\nTherefore, HRDE will behave as a phantom\nlike fluid even though the brane undergoes a de Sitter stage asymptotically. Fig. \\ref{Q}\nshows that the Universe\ncontinues accelerating in the future. For the model $Q=\\lambda_HH\\rho_H+\\lambda_m\\rho_m$, the sets of the couple\n$(\\lambda_m,\\lambda_H)$ that verifie the constraint $\\frac{3\\lambda_H}{3-\\lambda_m}=0.1$ lead to the same conclusions.\nWe conclude that this kind of interaction can describe the current acceleration expansion of our Universe.\n\n\\section{Model with Gauss-Bonnet term in the bulk}\n\nNow we consider the model where the bulk action contains a GB\ncurvature term in order to analyze the possibility of avoiding the big\nfreeze present in the absence of interaction between CDM and HRDE and in order\nto improve the constraints between the interaction and the beta parameters. In the same manner\nwe show that the avoidance of the big rip and little rip requires\nfinite values of $\\Omega_{\\mathrm{m}}$ at the far future.\n\nFrom Eq. (\\ref{variation of Omega m}), the solution for the interaction $\\lambda_{\\mathrm{m}}=3$ and $\\lambda_{H}=0$\nis similar to the case without the GB term which is already analysed in the case $Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$. Therefore,\nEq. (\\ref{variation of Omega m}) should be analyzed only for $\\lambda_{\\mathrm{m}}\\neq 3$ and can be\nwritten as\n\n\\begin{eqnarray}\n \\Big[ \\lambda_{H}\\beta -\\beta (\\lambda_{m}-3)\\Big] \\dfrac{dE}{dx}\n&=&- \\Big[ 2\\lambda_{H}\\beta -\\left( 2\\beta -1\\right) (\\lambda_{m}-3)%\n\\Big] E \\nonumber\\\\\n-&2&\\sqrt{\\Omega_{rc}}(\\lambda_{m}-3)(1+\\Omega_{\\alpha }E^{2})\n\\label{diff eq with GB}\n\\end{eqnarray}\nin the asymptotic regime where $d\\Omega_m\/dx\\rightarrow 0$.\\par\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth]{FIG4.pdf}\n\\end{center}\n\\caption{{Plot of the equation of state $\\omega_H$ against $x$ for $%\n\\lambda_{H}=0.1$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$. The plot has been done for several\nvalues of the parameter $\\beta$ that are stated at the bottom of the\nfigure.}}\n\\label{w}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{weff.pdf}\n\\end{center}\n\\caption{{Plot of the effective equation of state $\\omega_{\\text{eff}}$ against $x$ for $%\n\\lambda_{H}=0.1$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$. The plot has been done for several\nvalues of the parameter $\\beta$ that are stated at the bottom of the\nfigure.}}\n\\label{weff}\n\\end{figure}\nFor $\\lambda_{\\mathrm{m}}\\neq 3$, and in order to compare our results to\nthe case without interaction \\cite{OualiPRD85}, it is interesting to rewrite\nequation ({\\ref{diff eq with GB}) as}\n\n{%\n\\begin{equation}\n\\tilde{\\beta} \\dfrac{dE}{dx}=-\\left[ {\\left( 2\\tilde{\\beta}-1\\right)\nE-2\\sqrt{\\Omega_{rc}}(1+\\Omega_{\\alpha }E^{2})}\\right] ,\n\\label{FriedasympGB}\n\\end{equation}}\n\nwhere $\\tilde{\\beta} =\\gamma \\beta ,$ and\n\\begin{equation}\n\\ \\gamma =\\left(1+\\frac{\\lambda_{H}}{3-\\lambda_{m}}\\right). \\label{gamma}\n\\end{equation}\n\nThe solutions of Eq. (\\ref{FriedasympGB}) depends on the sign of the\ndiscriminant $\\mathcal{D}$ of the polynomial on its rhs, which\nreads\n\\begin{equation}\n\\mathcal{D}=\\left( 2\\tilde{\\beta}-1\\right) ^{2}-16\\Omega_{r_{c}}\\Omega\n_{\\alpha }, \\label{D}\n\\end{equation}%\n and can be factorised as follows\n\\begin{equation}\n\\mathcal{D}=\\mathcal{F}(\\beta -\\beta_{-})(\\beta -\\beta_{+}),\n\\label{factorD}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\n\\beta_{\\pm }=\\frac{1+\\Omega_{\\alpha }\\pm 2\\sqrt{\\Omega_{\\alpha }}%\n(1-\\Omega_{m_0})}{2[(1+\\Omega_{\\alpha })\\gamma \\pm \\sqrt{\\Omega_{\\alpha }}%\n(1-q_{0})]}.\\label{betapm}\n\\end{equation}\n\nand\n\n\\begin{equation}\n\\mathcal{F}=4\\left[\\gamma ^{2}-\\Omega_{\\alpha }\\left(\\frac{1-q_{0}}{1+\\Omega_{\\alpha\n}}\\right)^{2}\\right]. \\label{coeffF}\n\\end{equation}\nFig. \\ref{beta_pm} shows the behaviour of the parameters $\\beta_\\pm$ against $\\Omega_\\alpha$\nfor the cosmological parameters $\\Omega_{m0}=0.315$ and $q_0=-0.558$.\nAs can be seen from Fig. \\ref{beta_pm}, $\\beta_+=\\beta_-=\\beta_{\\textrm{lim}}$ for $\\gamma=\\frac{1}{2\\beta_{\\textrm{lim}}}$.\nFor $\\frac{1}{2\\beta_{\\textrm{lim}}}<\\gamma$ ($\\gamma<\\frac{1}{2\\beta_{\\textrm{lim}}}$), $\\beta_+<\\beta_-<\\beta_{\\textrm{lim}}$\n($\\beta_{\\textrm{lim}}<\\beta_+<\\beta_-$). \\par\nIn the following we analyze the effect of the interaction between CDM and HRDE component on the\nsingularities appearing on the same model without such interaction (cf. Ref. \\cite{OualiPRD85}).\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{Q.pdf}\n\\end{center}\n\\caption{{Plot of the deceleration parameter $q$ against $x$ for $%\n\\lambda_{H}=0.1$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$. The plot has been done for several values of\nthe parameter $\\beta$ that are stated at the bottom of the figure.}}\n\\label{Q}\n\\end{figure}\n\n\\subsection{$Q=\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$}\n\nIn this case and from Eq. (\\ref{gamma}) $\\gamma =1$, we obtain the\nsame asymptotic result as in Ref. \\cite{OualiPRD85}, and we conclude, as in\nthe previous section, that the IHRDE model with $Q=\\lambda_{\\mathrm{m%\n}}H\\rho_{\\mathrm{m}}$ does not succeed to remove the big freeze presented in\nthe range $\\beta_{\\lim }<\\beta <\\beta_{-}$ for the holographic Ricci dark\nenergy in a DGP brane world model with a GB term in the bulk.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG6.pdf}\n\\end{center}\n\\caption{{Plot of the $\\beta_\\pm$ parameters against $\\Omega_%\n\\alpha$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$. The plot has been done for several values of the parameter $%\n\\gamma$.}}\n\\label{beta_pm}\n\\end{figure}\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=0.75\\columnwidth]{FIG7.pdf}\n \\end{center}\n \\caption{{Plot of the dimensionless Hubble rate $E$ against $x$\n for an interacting coefficient $\\gamma =\\frac{1}{2%\n \\beta_{\\lim }}$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$ and $\\Omega_{\\alpha }=0.1$.\n The plot has been done for several values of the holographic parameter that\n are stated at the bottom of the figure.}}\n \\label{gamma0}\n \\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG8.pdf}\n\\end{center}\n\\caption{Plot of the dimensionless Hubble rate $E$ against $x$ for the\nnormal branch, and for the negative sign of $\\mathcal{D}$. The cosmological parameters\nare assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{%\n\\alpha }=0.1$, $\\lambda_{m}=0$ and $\\lambda_{H}=0.3$. The\nplot has been done for several values of the holographic parameter that are\nstated at the bottom of the figure.}\n\\label{Dn}\n\\end{figure}\n\\subsection{$Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$ and\n$Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}+\\lambda %\n_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$}\nThe analysis will be done for the model $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$ while\nthe generalization to the model $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}+\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$,\nas it can be noticed from Eq. (\\ref{diff eq}),\nwill be obtained by replacing $\\lambda_H$ by $\\frac{3\\lambda_H}{3-\\lambda_m}$\n\\subsubsection{$\\gamma =\\frac{1}{2\\beta_{\\lim }}$}\n\n\nIn this case $\\beta_{\\lim }=\\beta_-=\\beta_+$ and the discriminant is equal to $\\mathcal{D=}$ $\\mathcal{F}(\\beta\n-\\beta_{\\lim })^{2}$ which is always positive. The solution of the\nasymptotic Friedmann equation (\\ref{FriedasympGB}) reads:\n\\begin{equation}\n\\left\\vert \\frac{4\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}E\n-2\\gamma \\beta +1+\\sqrt{\\mathcal{D}}}{4\\Omega_{\\alpha }\\sqrt{%\n\\Omega_{r_{c}}}E-2\\gamma \\beta +1-\\sqrt{\\mathcal{D}}}\\right\\vert =%\n\\mathcal{C}_{2}\\exp \\left[ -\\frac{\\sqrt{\\mathcal{D}}}{\\gamma \\beta }(x-x_{2})%\n\\right] ,\n\\end{equation}%\nwhere\n\\begin{equation}\n\\mathcal{C}_{2}=\\left\\vert \\frac{4\\Omega_{\\alpha }\\sqrt{\\Omega\n_{r_{c}}}E_{2}-2\\gamma \\beta +1+\\sqrt{\\mathcal{D}}}{4\\Omega\n_{\\alpha }\\sqrt{\\Omega_{r_{c}}}E_{2}-2\\gamma \\beta +1-\\sqrt{\\mathcal{D}}}%\n\\right\\vert ,\n\\end{equation}%\n$x_{2}$ and $E_{2}$ are integration constants. We can deduce that, for\nvery large values of $x$, the brane behaves asymptotically like an expanding de Sitter\nuniverse, i.e. with a constant positive Hubble rate (see Eq. (\\ref{D}))\n\\begin{equation}\nE_{+}=\\frac{(2\\gamma \\beta -1)-\\sqrt{\\mathcal{D}}}{4\\Omega_{\\alpha }\\sqrt{%\n\\Omega_{r_{c}}}}=\\frac{(\\frac{\\beta }{\\beta_{\\lim }}-1)-\\sqrt{\\mathcal{D}}%\n}{4\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}}.\n\\end{equation}\n\n\\subsubsection{$\\gamma \\neq \\frac{1}{2\\beta_{\\lim }}$}\n\\begin{enumerate}\n\t\\item {Negative discriminant ($\\mathcal{D}<0$):} \nThe discriminant $\\mathcal{D}$ is negative for $\\frac{1}{2\\beta_{\\lim}}<\\gamma$\nwhere $\\beta $ satisfies $\\beta_{-}<\\beta <\\beta_{+}<\\beta_{\\lim }$ or for $\\gamma <\\frac{1}{2\\beta_{\\lim }}$\nwhere $\\beta_{\\lim }<\\beta_{+}<\\beta <\\beta_{-}$ (cf. Eq. (\\ref{factorD}) and Fig. \\ref{beta_pm}). The\nfirst case will be ignored since it corresponds to the self-accelerating\nbranch. Hence only the case $\\gamma <\\frac{1}{2\\beta_{\\lim }}$ will be\nanalysed. The Friedmann Eq. (\\ref{FriedasympGB}) can be integrated as\n\n\\begin{equation}\nE=\\frac{1}{4\\Omega_{\\alpha }\\sqrt{\\Omega_{rc}}}\\left\\{ \\sqrt{\\left\\vert\n\\mathcal{D}\\right\\vert }\\tan \\left[ \\frac{\\sqrt{\\left\\vert \\mathcal{D}%\n\\right\\vert }}{2\\gamma \\beta }\\left( x-\\mathcal{C}_{2}\\right) \\right]\n+2\\gamma \\beta -1\\right\\} , \\label{solution D<0}\n\\end{equation}%\nwhere\n\\begin{equation}\n\\mathcal{C}_{2}=x_{2}+\\frac{2\\gamma \\beta }{\\sqrt{\\left\\vert \\mathcal{D}%\n\\right\\vert }}\\arctan \\left[ \\frac{-4\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}%\nE_{2}+2\\gamma \\beta -1}{\\sqrt{\\left\\vert \\mathcal{D}\\right\\vert }}\\right] ,\n\\end{equation}%\n$x_{2}$ and $E_{2}$ are integration constants. Consequently, there is\nalways a finite value of the scale factor or $x$ {where the Hubble rate and its derivative blow\nup at}\n\\begin{equation}\nx_{\\mathrm{sing_{1}}}=\\mathcal{C}_{2}+\\frac{2\\gamma \\beta }{\\sqrt{\\left\\vert\n\\mathcal{D}\\right\\vert }}\\left(n+\\frac{1}{2}\\right)\\pi ,\\quad n\\in \\mathbb{Z},\n\\end{equation}%\nwhere $\"{\\mathrm{sing_{1}}}\"$ denotes the big freeze singularity.\nTherefore we conclude that the brane hits a big freeze singularity in the\nfuture as the event $x_{\\mathrm{sing_{1}}}$ takes place at a finite future\ncosmic time $t_{\\mathrm{sing_{1}}}$ \\cite{Noj,BouhmadiLopez:PLB2008}.\\\\\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG9.pdf}\n\\end{center}\n\\caption{Plot of the dimensionless Hubble rate $E$ against $x$, for the\npositive sign of $\\mathcal{D}$, and for an interacting coefficient such that $\\gamma <%\n\\frac{1}{2\\beta_{\\lim }} $. The cosmological parameters are assumed\nto be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$,\n$\\lambda_{m}=0$ and $\\lambda_{H}=0.3$. The plot has been\ndone for several values of the holographic parameter that are stated at the\nbottom of the figure.}\n\\label{Dp}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{FIG10.pdf}\n\\end{center}\n\\caption{Plot of the dimensionless Hubble rate $E$ against $x$, for the\npositive sign of $\\mathcal{D}$, and for an interacting coefficient such that $\\gamma >%\n\\frac{1}{2\\beta_{\\lim }} $. The cosmological parameters are assumed\nto be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$,\n$\\lambda_{m}=0$ and $\\lambda_{H}=0.6$. The plot has been\ndone for several values of the holographic parameter that are stated at the\nbottom of the figure.}\n\\label{Dp1}\n\\end{figure}\n\n\\item {Positive discriminant ($0<\\mathcal{D}$):}\n\n\nThe discriminant $\\mathcal{D}$ is positive for $\\frac{1}{2\\beta_{\\lim }}<\\gamma$,\n and $\\beta_{+}<\\beta_{\\lim }<\\beta $ or for $\\gamma <\\frac{1}{2\\beta_{\\lim }}$ \n and $\\beta $ satisfying $\\beta_{-}<\\beta $ or $\\beta_{\\lim }<\\beta <\\beta_{+}$\n(cf. Eq.~(\\ref{factorD}), Fig. \\ref{beta_pm}, see also Fig.~\\ref{Dp}, and \\ref{Dp1} which will be discussed later).\n\nThe solution of the asymptotic Friedmann equation (\\ref{FriedasympGB})\nreads\n\\begin{equation}\n\\left\\vert \\frac{4\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}%\nE-2\\gamma \\beta +1+\\sqrt{\\mathcal{D}}}{4\\Omega_{\\alpha }\\sqrt{%\n\\Omega_{r_{c}}}E-2\\gamma \\beta +1-\\sqrt{\\mathcal{D}}}\\right\\vert =%\n\\mathcal{C}_{2}\\exp \\left[ -\\frac{\\sqrt{\\mathcal{D}}}{\\gamma \\beta }(x-x_{2})%\n\\right] , \\label{solution D>0}\n\\end{equation}%\nwhere\n\\begin{equation}\n\\mathcal{C}_{2}=\\left\\vert \\frac{4\\Omega_{\\alpha }\\sqrt{\\Omega\n_{r_{c}}}E_{2}-2\\gamma \\beta +1+\\sqrt{\\mathcal{D}}}{4\\Omega\n_{\\alpha }\\sqrt{\\Omega_{r_{c}}}E_{2}-2\\gamma \\beta +1-\\sqrt{\\mathcal{D}}}%\n\\right\\vert ,\n\\end{equation}%\n$x_{2}$ and $E_{2}$ are integration constants. We can deduce that the brane\nbehaves asymptotically like an expanding de Sitter universe with a positive Hubble rate\n\\begin{equation}\nE_{+}=\\frac{(2\\gamma \\beta -1)-\\sqrt{\\mathcal{D}}}{4\\Omega_{\\alpha }\\sqrt{%\n\\Omega_{r_{c}}}}, \\label{E_{+2}}\n\\end{equation}%\n for $\\gamma >\\frac{1}{2\\beta }$. While for the\ncase, $\\gamma \\leq \\frac{1}{2\\beta }$ the Hubble rate $E_{+}$ is negative\nand the asymptotic analysis performed is no longer valid. In fact, the brane\nfaces a big freeze singularity where, from Eq. (\\ref{diff eq with GB}), the expansion of the brane can be\napproximated by\n\\begin{equation}\nE\\sim \\frac{\\gamma\\beta}{2\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}\\,(x_{\\mathrm{sing_{2}%\n}}-x)}. \\label{solution2 D>0}\n\\end{equation}%\nThe constant $x_{\\mathrm{sing_{2}}}$ stands for the \\textquotedblleft\nsize\\textquotedblright\\ of the brane at this big freeze singularity $\"{\\mathrm{sing_{2}}}\"$. The\nHubble rate Eq. (\\ref{solution2 D>0}) can be integrated over time, resulting in\nthe following expansion for the scale factor of the brane\n\\begin{equation}\na=a_{\\mathrm{sing_{2}}}\\exp \\left[ -\\left( \\frac{H_{0}}{\\Omega_{\\alpha }%\n\\sqrt{\\Omega_{r_{c}}}}\\right) ^{\\frac{1}{2}}\\sqrt{t_{\\mathrm{sing_{2}}}-t}%\n\\right].\n\\end{equation}%\nThe big freeze singularity takes place at a finite scale factor, $a_{\\mathrm{%\nsing_{2}}}$, and a finite cosmic time, $t_{\\mathrm{sing_{2}}}$. This latter\ncase can be removed by an appropriate choice of $\\lambda_{m}$ and $\\lambda\n_{H}$ through the coupling $\\gamma$, which allows removing the big freeze present in the\ncase $\\mathcal{D>}0$ by making the brane asymptotically de Sitter.\\\\\n\n\\item {Vanishing discriminant ($\\mathcal{D}=0$):}\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{fig11.pdf}\n\\end{center}\n\\caption{{}Plot of the dimensionless Hubble rate $E$ against $x$, for\nvanishing $\\mathcal{D}$, and for an interacting coefficient $\\gamma <\\frac{1}{2%\n\\beta_{\\lim }}$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$, and $%\n\\lambda_{m}=0$. The plot has been done for several values of the\nholographic parameter, and the interacting parameter $\\lambda_{H}$ that\nare stated at the bottom of the figure.}\n\\label{D=0}\n\\end{figure}\nFinally, the discriminant $\\mathcal{D}$ vanishes when $\\beta =\\beta_{+}$ or\n$\\beta_{-}$. This can take place on the normal branch for $\\gamma <\\frac{1}{%\n2\\beta_{\\lim }}$. The solution of the modified Friedmann equation (\\ref%\n{FriedasympGB}) can be expressed as\n\\begin{equation}\nE=-\\frac{1}{4\\Omega_{\\alpha }\\sqrt{\\Omega_{r_{c}}}}\\left[ (2\\gamma \\beta\n-1)-\\frac{1}{\\mathcal{C}_{3}+\\frac{1}{2\\gamma \\beta }(x-x_{3})}\\right] ,\n\\label{solution2 D=0}\n\\end{equation}%\nwhere\n\\begin{equation}\n\\mathcal{C}_{3}=\\frac{1}{2\\gamma \\beta -1+4\\Omega_{\\alpha }\\sqrt{\\Omega\n_{r_{c}}}E_{3}},\n\\end{equation}%\n$x_{3}$ and $E_{3}$ are integration constants. By taking the limit $%\nx\\rightarrow \\infty $, the dimensionless Hubble rate (\\ref{solution2 D=0})\nreduces to\n\\begin{equation}\nE_{+}=\\frac{(2\\gamma \\beta -1)}{4\\Omega_{\\alpha }\\sqrt{\\Omega\n_{r_{c}}}}, \\label{E_{+3}}\n\\end{equation}%\n$E_{+}$ coincides with the one of Eq.~(\\ref{E_{+2}}) for a vanishing $%\n\\mathcal{D}$. The brane can be asymptotically de Sitter for the normal\nbranch if $\\frac{1}{2\\beta }<\\gamma <\\frac{1}{2\\beta_{\\lim }}$, otherwise\nthe Hubble rate $E_{+}$ is negative and the asymptotic analysis performed is no longer valid.\nThis behavior can be seen from the numerical analysis presented in Fig. {\\ref{D=0}} and the brane undergoes a big freeze\nsingularity with an expansion given by Eq. (\\ref{solution2 D>0}).\n\nThe results of the model for $Q=\\lambda_{%\n\\mathrm{H}}H\\rho_{\\mathrm{H}}+\\lambda_{\\mathrm{m}}H\\rho_{\\mathrm{m}}$ are\nsimilar to those with $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$ after making the\ntransformation $\\lambda_{H}$ $\\longrightarrow \\frac{3\\lambda_{H}}{%\n3-\\lambda_{m}}$ in the model $Q=\\lambda_{\\mathrm{H}}H\\rho_{\\mathrm{H}}$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{betalim}\n\\end{center}\n\\caption{Plot of the effective equation of state $\\omega_{eff}$ against $x$, for $\\gamma =\\frac{1}{%\n2\\beta_{\\lim }}$. The cosmological parameters are assumed to be $\\Omega\n_{m_{0}}=0.315$, $q_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$. The plot has been\ndone for several values of the holographic parameter that are stated at the\nbottom of the figure.}\n\\label{betalim}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{Dnegative}\n\\end{center}\n\\caption{Plot of the effective equation of state $\\omega_{eff}$ against $x$, for the negative sign of\n$\\mathcal{D}$, which correspond to $\\gamma <\\frac{1}{2\\beta_{\\lim }}$. The\ncosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $q_{0}=-0.558\n$, $\\Omega_{\\alpha }=0.1$, $\\lambda_{m}=0$ and $\\lambda_{H}=0.3$. The\nplot has been done for several values of the holographic parameter that are\nstated at the bottom of the figure.}\n\\label{Dnegative}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{Dnul}\n\\end{center}\n\\caption{Plot of the effective equation of state $\\omega_{eff}$ against $x$, for the vanishing $\\mathcal{D}$.\nThe cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$, $\\lambda_{m}=0$. The plot has been\ndone for several values of the holographic parameter that are stated at the\nbottom of the figure.}\n\\label{D0}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{Dpositive}\n\\end{center}\n\\caption{Plot of the effective equation of state $\\omega_{eff}$ against $x$, for the positive sign of\n$\\mathcal{D}$. The cosmological parameters are assumed to be $\\Omega_{m_{0}}=0.315$, $%\nq_{0}=-0.558$, $\\Omega_{\\alpha }=0.1$, $\\lambda_{m}=0$. The plot has been\ndone for several values of the holographic parameter, and the interacting parameter $\\lambda_{H}$\nthat are stated at the\nbottom of the figure.}\n\\label{Dpositive}\n\\end{figure}\n\n\\end{enumerate}\n\\subsection{Numerical analysis}\n\nThe asymptotical analysis we have carried out in the previous subsection can\nbe completed with a numerical analysis of Eq. (\\ref{variation double de E}).\n\nFig. {\\ref{gamma0}} shows some numerical solutions for $\\gamma =%\n\\frac{1}{2\\beta_{\\lim }}$ corresponding to the normal branch in which we\nare interested. As we notice, the brane is asymptotically de Sitter in the\nfuture, and the dimensionless Hubble rate $E$ approaches the value\n$E_{+}=\\frac{(\\frac{\\beta }{\\beta_{\\lim }}-1)-\\sqrt{\\mathcal{D}}}{4\\Omega\n_{\\alpha }\\sqrt{\\Omega_{r_{c}}}}.$\nFor $\\gamma \\neq \\frac{1}{2\\beta_{\\lim }}$, we discuss the numerical\nanalysis with respect to the sign of the discriminant $\\mathcal{D}$:\n(i) Fig.~\\ref{Dn} shows the numerical solutions of Eq.\n(\\ref{variation double de E}) for a negative value of $\\mathcal{D}$.\nThe brane expands with respect to $x$ in the future until it reaches a big\nfreeze singularity which is consistent with our analytical results, (ii) For a\npositive sign of $\\mathcal{D}$ and for $\\gamma <\\frac{1}{2\\beta_{\\lim }}$,\nFig. \\ref{Dp} shows that the brane expands until it reaches a big\nfreeze singularity for $\\beta =0.45$, and $\\beta =0.52$ for $\\gamma <\\frac{1}{2\\beta }$,\nand a de Sitter solution for $\\beta =1\/2$ and $\\beta =0.6$,\nfor $\\gamma >\\frac{1}{2\\beta }$. These results are similar to\nour previous analytical analysis. For $\\gamma >\\frac{1}{2\\beta_{\\lim }}$\nthe numerical solutions behave like a de Sitter solutions as shown in Fig. \\ref{Dp1},\nwhich is of course consistent with the analytical results,\n(iii) When the discriminant $\\mathcal{D}$ vanishes, the solutions are shown in\nFig {\\ref{D=0}}. The brane expands with respect to $x$ in the future until\nit reaches a big freeze for $\\beta =0.45$, and $\\beta =0.46$ for\n$\\gamma <\\frac{1}{2\\beta }$, and behaves like a de Sitter solution for $\\beta =0.47$,\nand $\\beta =0.51$ for $\\gamma >\\frac{1}{2\\beta }$. All of these\nsolutions again are consistent with our previous analytical analysis.\\par\nFor completeness, we plot in Figs. \\ref{betalim}-\\ref{Dpositive} the behaviours of\nthe effective equation of state, $\\omega_{eff}$, defined in Eq. (\\ref{rhoeff1})-(\\ref{rhoeff2}) where\n$\\rho_{\\text{eff}}$ reads in this case\n\\begin{equation}\\label{}\n\\rho_{\\text{eff}}= 3M_p^2H_0^2\\Big[\\frac{1}{2}\\beta\\frac{dE^2}{dx}+2\\beta E^2 -2\\sqrt{\\Omega_{\\mathrm{r_c}}}(1+\\Omega_{\\mathrm{\\alpha}}E^{2})E\\Big]\n\\end{equation}\nFig. \\ref{betalim} shows the behaviours of\nthe effective equation of state for $\\gamma=\\frac{1}{2\\beta_{\\textrm{lim}}}$. At the far future $\\omega_{eff}$\nhas a phantom like even though the brane follows a de sitter behaviour.\\par\nFig. \\ref{Dnegative} shows the behaviours of\nthe effective equation of state for the negative sign of $\\mathcal{D}$ i.e. $\\gamma<\\frac{1}{2\\beta_{\\textrm{lim}}}$.\nThe parameter $\\omega_{eff}$ has a phantom like behaviour and the brane ends its expansion in a big freeze singularity.\\par\nFig. \\ref{D0} shows the behaviours of\nthe effective equation of state for the vanishing $\\mathcal{D}$.\nOnce again the parameter $\\omega_{eff}$ has a phantom like for $\\frac{1}{2\\beta}<\\gamma<\\frac{1}{2\\beta_{\\textrm{lim}}}$ even though the brane follows a de sitter expansion\nwhile for $\\gamma<\\frac{1}{2\\beta}<\\frac{1}{2\\beta_{\\textrm{lim}}}$ the brane ends its behaviours in a big freeze singularity.\\par\nFig. \\ref{Dpositive} shows the behaviours of\nthe effective equation of state for a positive sign of $\\mathcal{D}$.\nThe equation of state parameter $\\omega_{eff}$ has a phantom like behaviour with two possible end states of the brane: (i) A de sitter behaviour of the brane for $\\frac{1}{2\\beta_{\\textrm{lim}}}<\\gamma$ e.g.\n$\\beta=0.6$ and $\\gamma=0.6$ and for $\\frac{1}{2\\beta}<\\gamma<\\frac{1}{2\\beta_{\\textrm{lim}}}$ e.g. $\\beta=0.5$ and $\\gamma=1.1$ (ii) while for $\\gamma<\\frac{1}{2\\beta_{\\textrm{lim}}}$ the brane ends its behaviours in\na big freeze singularity.\n\n\n\n\\begin{center}\n\\begin{table*}[t!]\n \\centering\n \\begin{tabular}{ccccccc}\n \\toprule\n \n\t\t& & & Without a Gauss-Bonnet term & & & \\\\\n\t\t\\hline\\hline\n Sec. & Interacting model & ~ $\\lambda$~ & $\\beta$ & &Late time behaviour & \\\\ \\hline\\hline\n \n \n III A & $\\lambda_{\\rm m}\\neq 0, \\lambda_{\\rm H}=0$ & {$\\lambda_{\\rm m}=3$} & $-$ & &Non physical \\\\\n\n \n & & {$\\lambda_{\\rm m}\\neq 3$} & $\\beta=\\frac{1}{2}$ & & LR \\\\\n \n & & {$\\lambda_{\\rm m}\\neq 3$} & $\\beta>\\frac{1}{2}$ & & de Sitter \\\\\n \n & & {$\\lambda_{\\rm m}\\neq 3$} & $\\beta_{lim}<\\beta<\\frac{1}{2}$ & & BR \\\\\n \n\\hline\n \nIII B & $\\lambda_{\\rm m}=0, \\lambda_{\\rm H}\\neq 0$ &-$-$& $\\beta=\\beta_{LR}=\\frac{3}{2(\\lambda_{\\rm H}+3)}$& & LR \\\\\n \n & & & $\\beta<\\beta_{LR}$& & {BR} \\\\\n \n & & & $\\beta>\\beta_{LR}$& & {de Sitter} \\\\ \\hline\n \nIII C & $\\lambda_{\\rm m}\\neq 0, \\lambda_{\\rm H}\\neq 0$ & {$\\lambda_{\\rm m}=3$} & $-$ & & Minkowski \\\\\n \n & & {$\\lambda_{\\rm H}=\\lambda_{\\rm m}-3$} & $-$ & & Non physical \\\\\n \n & & {$\\lambda_{\\rm H}\\neq \\lambda_{\\rm m}-3$} & $\\beta=\\beta_{LR}=\\frac{3-\\lambda_{\\rm m}}{2(\\lambda_{\\rm H}-2(\\lambda_{\\rm m}+6)}$ & & LR \\\\\n\n & & {$$} & $\\beta>\\beta_{LR}$ & & de Sitter \\\\\n\n & & {$$} & $\\beta<\\beta_{LR}$ & & BR \\\\\n\n \\hline\\hline\n&\t\t& & With a Gauss-Bonnet term & & &\\\\\n\t\t\\hline\\hline\n Sec. & Interacting model & ~ ${\\cal{D}}$~ &$\\gamma$& $\\lambda$ & Late time behaviour & \\\\ \\hline\\hline\n \n \n IV A & $\\lambda_{\\rm m}\\neq 0, \\lambda_{\\rm H}=0$ & {$-$} & $1$ &$\\lambda_{\\rm m}=3 $& Non physical \\\\\n\n \n & & {$-$} & $1$ &$\\lambda_{\\rm m}\\neq3 $& Big Freeze \\cite{OualiPRD85} \\\\\n\n\\hline\n \nIV B $^*$ & $\\lambda_{\\rm m}= 0, \\lambda_{\\rm H}\\neq 0$ & {${\\cal{D}}<0$} & $\\gamma<\\frac{1}{2\\beta_{lim}}$ &$- $& Big Freeze\\\\\n\n \n \n\n\n & & {$0\\leq {\\cal{D}}$} & $\\frac{1}{2\\beta}<\\gamma\\leq\\frac{1}{2\\beta_{lim}}$ &$- $& de Sitter \\\\\n\n \n & & {$$} & $\\gamma<\\frac{1}{2\\beta}<\\frac{1}{2\\beta_{lim}}$ &$- $& Big Freeze \\\\\n \n \n \n \n\n\n \\hline\\hline\n \n \\end{tabular}\n \n \n \\caption{Summary of the behaviours of the universe at late time, for different DM and DE interactions. ($^*$) The case $\\lambda_{\\rm m}\\neq 0, \\lambda_{\\rm H}\\neq 0$ is obtained by replacing $\\lambda_{\\rm H}$ by $\\frac{3\\lambda_{\\rm H}}{3-\\lambda_{\\rm m}}$ in the expressions of $\\gamma$ and ${\\cal{D}}$. }\n \\label{Fate-Universe}\n\\end{table*}\n\\end{center}\n\n\n\n\\section{Conclusions}\n\nIn this paper, we have presented an interacting holographic Ricci dark\nenergy model (IHRDE) with a cold dark matter (CDM) component in an induced gravity\nbrane world model. We have shown that the late time acceleration of\nthe universe is consistent with the current observational data with and without a Gauss-Bonnet (GB) curvature term in the\nbulk action. \\\\\n\nThe parameter $\\beta $ that characterizes HRDE is very\nimportant in determining the asymptotic behaviour of the holographic Ricci\ndark energy (HRDE) and that of the brane. The parameter $\\beta$ is bounded by a limiting value, $%\n\\beta_{\\textrm{lim}}$, that splits the self-accelerating branch from the normal one.\nIn this paper, we have considered only the normal branch,\ni.e. $\\beta >\\beta_{\\textrm{lim}}$, which suffers from the big rip and the\nlittle rip singularities (see Ref. \\cite{OualiPRD85}).\nAssuming that, at present, our model does not deviate too much from $%\n\\Lambda $CDM, the value of $\\beta_{\\textrm{lim}}$ is estimated to be of the order $0.44$.\\\\\n\nThe interaction is characterized by the quantity $Q=\\lambda_HH\\rho_H+%\n\\lambda_mH\\rho_m$ where $\\lambda_H$ and $\\lambda_m$ are the coupling\ncharacterizing the HRDE and the CDM interaction.\\\\\n\nFor a vanishing $\\lambda_{H}$, the interaction model characterized only by the CDM energy does not succeed to remove\nthe big freeze singularity occurring in the non interacting model \\cite{OualiPRD85}\nwith and without the GB curvature term.\\\\\n\nFor a vanishing $\\lambda_{m}$, the model with interaction between the dark sector of\nthe universe can be splited into two cases:\n\n\\begin{itemize}\n\\item Without a GB term the IHRDE model shows that the interaction removes the\nlittle rip singularity for $\\beta =1\/2$ to $\\beta =%\n\\frac{3}{2(3+\\lambda_{H})}$, and reduces the width of the big rip\nsingularity from $\\beta_{\\textrm{lim}}<\\beta <1\/2$ to\n$\\beta_{\\textrm{lim}}<\\beta <\\frac{3}{2(3+\\lambda_{H})}<1\/2$. Therefore an appropriate choice of the\ncoupling $\\lambda_{H}$ such that $\\frac{3}{2(3+\\lambda_{H})}<\\beta\n_{lim}$ avoids the big rip and the little rip and hence the IHRDE gives\na satisfactory and an alternative description of\nthe late time cosmic acceleration of the universe as compared to the HRDE. Indeed the later\none modifies the big rip and little rip into a big freeze one\nwhile the former removes them definitively. Furthermore the IHRDE will have a\nphantom-like behaviour even though the brane undergoes a de Sitter stage at the very late time. \\\\\n\n\\item With a GB term in the bulk the IHRDE model depends on the sign of the discriminant ${\\mathcal{D}}$ through\nthe parameter $\\beta $, the GB parameter, and the coupling $\\gamma $.\nIn the particular case $\\gamma =\\frac{1}{2\\beta_{\\textrm{lim}}}$,\nthe interacting model succeed in removing the big rip and little\nrip singularity from the brane future evolution and it will evolve\nasymptotically as a de Sitter universe as is shown in Fig. \\ref{betalim}. For $\\gamma \\neq \\frac{1}{2\\beta_{\\textrm{lim}}}$, the situation depends on\nthe sign of the discriminant ${\\mathcal{D}}$:\n\n\\begin{enumerate}\n\\item If ${\\mathcal{D<}}0$, which correspond to $\\gamma <\\frac{1}{2\\beta\n_{\\lim }}$. The brane expand in the future until it reaches a big freeze\nsingularity as is shown in Fig. \\ref{Dnegative}.\n\n\\item If ${\\mathcal{D}}=0$, two situations can be found (as is shown in Fig. \\ref{D0})\n\n\\begin{description}\n\\item[a-] When $\\frac{1}{2\\beta }<\\gamma <\\frac{1}{2\\beta_{\\textrm{lim}}}$, the brane\nis asymptotically de Sitter.\n\n\\item[b-] When $\\gamma <\\frac{1}{2\\beta }<\\frac{1}{2\\beta_{\\textrm{lim}}}$, the brane\nhits a big freeze singularity.\n\\end{description}\n\n\\item If ${\\mathcal{D}}>0$, two situations can be found (as is shown in Fig. \\ref{Dpositive})\n\n\\begin{description}\n\\item[a-] When $\\gamma >\\frac{1}{2\\beta_{\\textrm{lim}}},$ the brane expands in the future\nuntil it reaches a de Sitter stage.\n\n\\item[b-] When $\\gamma <\\frac{1}{2\\beta_{\\textrm{lim}}},$\nthe brane becomes asymptotically de Sitter in the future for $\\gamma >\\frac{1}{2\\beta }$, while\nfor $\\gamma <\\frac{1}{2\\beta }$ the brane hits a big\nfreeze singularity.\n\\end{description}\n\\end{enumerate}\n\\end{itemize}\n\nTaking into account both couplings $\\lambda_m$ and $\\lambda_H$ the\ndescription of the interaction between CDM and HRDE is\nsimilar to the interaction characterised only by $\\lambda_H$ in the HRDE by\nreplacing the coupling $\\lambda_H$ in the interaction form $Q=\\lambda_HH\\rho_H$ by the\nquantity $\\frac{3\\lambda_H}{3-\\lambda_m}$. The same conclusions are obtained by requiring that the constraints\non the parameter $\\gamma$ are verified for the sets of the couple $(\\lambda_m,\\lambda_H)$ for which $\\gamma=1+\\frac{\\lambda_H}{3-\\lambda_m}$.\n\n\n\\acknowledgments\n\nThe research of M. B.-L. is supported by the Basque Foundation of Science Ikerbasque. She also would like to acknowledge the partial support from the Basque government Grant No. IT956-16 (Spain) and the project FIS2017-85076-P (MINECO\/AEI\/FEDER, UE).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}