Content-Type: multipart/mixed; boundary="-------------1101241108965" This is a multi-part message in MIME format. ---------------1101241108965 Content-Type: text/plain; name="11-10.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-10.comments" 8 pages ---------------1101241108965 Content-Type: text/plain; name="11-10.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-10.keywords" monomer-dimer, expansion ---------------1101241108965 Content-Type: application/x-tex; name="mdp.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mdp.tex" \documentclass[letterpaper,10pt]{article} \usepackage[utf8x]{inputenc} \usepackage{amsmath, amssymb, amsfonts} \usepackage{amsthm} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} %opening \title{Convergence of the Formal Expansion for $\lambda_d (p)$ of the Monomer-Dimer Problem for Small $p$} \author{Paul Federbush \\ Department of Mathematics \\ University of Michigan \\ Ann Arbor, MI 48109-1043 \\ (pfed@umich.edu)} \begin{document} \maketitle \begin{abstract} Shmuel Friedland and the author recently presented a formal expansion for $\lambda_d (p)$ of the monomer-dimer problem. Herein we prove that if the terms in the expansion are rearranged as a power series in $p$, then for sufficiently small $p$ this series converges. \end{abstract} In a series of papers the author presented a formal asymptotic expansion for $\lambda_d$ of the dimer problem, in inverse powers of $d$. See \cite{FedComp}. The expansion is as follows \begin{align} \lambda_d \sim \frac{1}{2} \ln (2d) - \frac{1}{2} + \sum_{k=1} \frac{c_k}{d^k} \end{align} computed through the $k=3$ term as \begin{align} \lambda_d \sim \frac{1}{2} \ln (2d) - \frac{1}{2} + \frac{1}{8} \frac{1}{d} + \frac{5}{96} \frac{1}{d^2} + \frac{5}{64} \frac{1}{d^3}. \end{align} In a recent paper, \cite{FedFried}, Shmuel Friedland and the author extended this work to yield a formal asymptotic expansion for $\lambda_d (p)$ of the dimer-monomer problem \begin{align} \label{FedFriedExpand} \lambda_d (p) \sim \frac{1}{2} \left( p \ln (2d) - p \ln p - 2 (1-p) \ln (1-p) - p \right) + \sum_{k=1} \frac{c_k (p)}{d^k} \end{align} computed through the $k=3$ term as \begin{align} \label{FedFriedk3} \lambda_d (p) \sim & \frac{1}{2} \left( p \ln (2d) - p \ln p - 2 (1-p) \ln (1-p) - p \right) \notag \\ & + \frac{1}{8} \frac{p^2}{d} + \frac{\left( 2p^3 +3p^4 \right)}{96} \frac{1}{d^2} + \frac{\left( -5 p^4 + 12 p^5 + 8 p^6 \right)}{192} \frac{1}{d^3}. \end{align} For given $d$ we rearrange the expansion in \eqref{FedFriedExpand} as a power series in $p$ \begin{align} \label{ArrangePowerSeries} \lambda_d (p) \sim \frac{1}{2} \left( p \ln (2d) - p \ln p - 2 (1-p) \ln (1-p) - p \right) + \sum_{k=2} a_{d,k} p^k. \end{align} We see from \eqref{FedFriedk3} that \begin{align} a_{d,2} &= \frac{1}{8} \frac{1}{d} \\ a_{d,3} &= \frac{1}{48} \frac{1}{d^2} \\ a_{d,4} &= \frac{1}{32} \frac{1}{d^2} - \frac{5}{192} \frac{1}{d^3}. \end{align} We have here used the fact that $C_k (p)$ of equation \eqref{FedFriedExpand} is a sum of powers $p^s$ where $k < s \le 2k$, see Lemma \ref{lemma4} and Theorem \ref{theorem_absconv} below. It is the primary goal of this paper to show that if $p$ is small enough ($0 \le p < p_0$, $p_0$ independent of $d$) the sum in \eqref{ArrangePowerSeries} converges, see Theorem \ref{theorem_p0val} below. (Throughout the paper we are not careful about getting the best value of $p_0$; with any improvements we could make to the current procedure the value we get for $p_0$ would still be anemic.) We will assume familiarity with Section 5 of \cite{FedFried}, and use many of the formulae therefrom. $\lambda_d (p)$ is determined, by a complicated computation, from the infinite sequence of cluster expansion kernels \begin{align} \bar J_1, \bar J_2, \bar J_3 , \dots \end{align} defined in equations (5.21), (5.23). (We will not indicate herein that such (5.--) equation comes from \cite{FedFried}.) The first six $\bar J_i$ have been computed and are listed in (5.25) -- (5.30). From (5.17) and (5.31) an infinite sequence of auxiliary quantities \begin{align} \label{alphas} \alpha_1 , \alpha_2 , \dots \end{align} are computed from the $\bar J_i$. An easy computation from (5.17) and (5.31) leads to the nice expression \begin{align} \label{Master1} \alpha_k = \left( \bar J_k p^k \right) \cdot \frac{1}{\left( 1 - 2 \sum i \alpha_i \right)^{2k}} \cdot \left( 1 - 2 \sum i \alpha_i / p \right)^{k} \end{align} which replaces (5.31). We view the $\alpha_k$ as determined from \eqref{Master1} by recursive iteration. Later working with bounds on the $\bar J_k$ we will study values of $p$ for which iterations converge to a \emph{solution} of \eqref{Master1}. From (5.10), (5.11), and (5.12) we have that \begin{align} \lambda_d (p) = S + \lim_{N \to \infty} \frac{1}{N} \ln Z^* \end{align} where we have defined \begin{align} \label{Master2} S \equiv \frac{p}{2} \ln (2d) - \frac{p}{2} \ln p - (1-p) \ln (1-p) - \frac{p}{2}. \end{align} Now from (5.32), (5.31), and (5.17) we may easily compute \begin{align} \label{Master3} \lambda_d (p) = S + \sum \alpha_i - \sum_{k=2} \frac{1}{k} \left( 2 \sum_i i \alpha_i \right)^k + \frac{1}{2} p \sum_{k=2} \frac{1}{k} \left( 2 \sum_i i \alpha_i / p \right)^k. \end{align} Equations \eqref{Master1}, \eqref{Master2}, and \eqref{Master3} are our master equations. All our results below concern solutions of these equations, we do not address here whether such solutions actually correspond to a computation of the monomer-dimer partition function as \begin{align} \sum \mathrm{covers} \sim e^{N \lambda_d (p)} \end{align} although certainly this is the case. We state the information in equation (5.22) as a lemma. \begin{lemma} $\bar J_k$ is a sum of inverse powers of $d$, $\left( 1/d \right)^s$, with \begin{align} \frac{k}{2} \le s < k \end{align} \end{lemma} \begin{lemma} At the first iteration of equation \eqref{Master1} $\alpha_k$ is a sum of powers of $p$ and $\left( 1 / d \right)$, $p^i \left( 1 / d \right)^j$, with \begin{align} i &= k \notag \\ \frac{i}{2} \le j & < i \end{align} \end{lemma} \begin{lemma} At the end of any number of iterations of equation \eqref{Master1} $\alpha_k$ is a sum of terms $p^i \left( 1 / d \right)^j$ with \begin{align} \label{ItSum} i & \ge k \notag \\ \frac{i}{2} \le j & < i \end{align} \end{lemma} \begin{lemma} \label{lemma4} Substituting the $\alpha_k$ as satisfying \eqref{ItSum} into \eqref{Master3} one finds $\lambda_d (p) - S$ is a sum of terms $p^i \left( 1 / d \right)^j$ satisfying \eqref{ItSum}. \end{lemma} These lemmas are easily proven by studying the evolution of powers of $p$ and $\left( 1 / d \right)$ through the iterations and expansions. One may consider the formal expansion of $\alpha_k$ after an infinite number of iterations of \eqref{Master1}, and its substitution into \eqref{Master3}, yielding an infinite formal expansion for $\lambda_d (p) - S$. These also are a sum of terms $p^i \left( 1 / d \right)^j$ satisfying \eqref{ItSum}. We reorganize our formal expansions as a power series in $p$. \begin{align} \alpha_k &= \sum_{s=k} p^s f_{k,s} \\ \lambda_d (p) &= S + \sum_{s=2} p^s g_s \end{align} The $f_{k,s}$ and $g_s$ are built up of powers of $\left( 1 / d \right)$, $ \left( 1 / d \right)^i$ satisfying \begin{align} \frac{s}{2} \le i < s \end{align} We now consider working with a fixed value of $d$, and assume we have a bound on the $\bar J_k$ \begin{align} \label{Jbarbound} \lvert \bar J_k \rvert \le B^k, \quad k = 1,2, \dots \end{align} for some $B$. Under these circumstances we set up the machinery to use the contraction mapping principle. On any formal infinite polynomial in $p$ \begin{align} f = \sum a_i p^i \end{align} we define a norm $\lvert f \rvert$ \begin{align} \lvert f \rvert \equiv \sum \lvert a_i p^i \rvert . \end{align} This norm has the properties \begin{align} &P1) & \lvert cf \rvert &= \lvert c \rvert \lvert f \rvert \label{P1} \\ &P2) & \lvert f + g \rvert & \le \lvert f \rvert + \lvert g \rvert \label{P2} \\ &P3) & \lvert f g \rvert & \le \lvert f \rvert \lvert g \rvert \label{P3} \end{align} for scalar $c$ and polynomials $f$ and $g$. We denote the sequence of $\alpha_k$, as in \eqref{alphas}, by $\alpha$, and define a norm on $\alpha$ \begin{align} \label{defnorm} \| \alpha \| = \sum_{k} 2^k \lvert \alpha_k \rvert . \end{align} We find an $\varepsilon$, $0 < \varepsilon < 1 / 2$, small enough so that \begin{align} \label{epssmallcond} \frac{1}{2} \frac{1}{ \left( 1 - 2 \varepsilon \right)^2} \left( 1 + 2 \varepsilon \right) \le 1 \end{align} and \begin{align} \label{contractcond} \frac{6 \varepsilon}{1 - 2 \varepsilon} \le 1. \end{align} We then require $p > 0$ to be small enough that \begin{align} \label{psmallcond} p^{k-1} B^k \le \varepsilon \frac{1}{8^k}, \quad k = 2,3, \dots \end{align} Working with this choice of $\varepsilon$ and $p$ we define the complete metric space $\mathcal{S}$ on which we establish a contraction mapping \begin{align} \mathcal{S} = \left\{ \alpha = \left\{ \alpha_k \right\} \vert \ \| \alpha \| \le p \varepsilon \right\} \end{align} We rewrite \eqref{Master1} as \begin{align} \label{alphakeq} \alpha_k = f_k \left( \alpha \right), \quad k = 2, 3, \dots \end{align} or \begin{align} \label{alphaeq} \alpha = f \left( \alpha \right). \end{align} Conditions \eqref{epssmallcond} and \eqref{psmallcond} ensure that $f$ carries $\mathcal{S}$ into $\mathcal{S}$. With the further condition \eqref{contractcond} one establishes that $f$ is a contraction. \begin{theorem} With the conditions on $p$ and $\varepsilon$ above, there is a unique solution of \eqref{alphaeq} in $\mathcal{S}$, exactly the one obtained by iteration of \eqref{Master1}. \end{theorem} Substituting this solution into \eqref{Master3} one obtains the expression for $\lambda_d (p)$. We collect the properties of this quantity. \begin{theorem} \label{theorem_absconv} For $0 < p \le p_0$, $p_0$ determined by \eqref{psmallcond}, \begin{align} \label{lambdasum} \lambda_d (p) = \frac{p}{2} \ln (2d) - \frac{p}{2} \ln p - (1-p) \ln (1-p) - \frac{p}{2} + \sum_{s=2} p^s g_s \end{align} where $g_s$ is a polynomial in $\left( 1 / d \right)$ with powers $\left( 1 / d \right)^i$ satisfying \begin{align} \frac{s}{2} \le i < s. \end{align} The sum in \eqref{lambdasum} is absolutely convergent. $g_s$ is a polynomial in $\bar J_1 , \bar J_2 , \dots , \bar J_s$ and is determined by a finite number of iterations of \eqref{Master1} substituted into \eqref{Master3}. One need only keep the finite number of terms throughout whose power of $p$ is less than or equal to $s$ to get $g_s$. \end{theorem} We content ourselves with presenting the proof that the $f$ of \eqref{alphaeq} maps $\mathcal{S}$ into $\mathcal{S}$. We look at the mapping of \eqref{alphakeq} carrying $\alpha_k$ into $\alpha'_k$ \begin{align} \alpha'_k = f_k \left( \alpha \right) \end{align} and we wish to prove if $\alpha$ is in $\mathcal{S}$ then $\alpha'$ is in $\mathcal{S}$. Parallel to \eqref{Master1} we have \begin{align} \alpha'_k = \left( \bar J_k p^k \right) \cdot \frac{1}{\left( 1 - 2 \sum i \alpha_i \right)^{2k}} \left( 1 - 2 \sum i \alpha_i / p \right)^k. \end{align} We take the $\lvert \cdot \rvert$ norm of both sides using \emph{P1, P2, P3} of \eqref{P1}--\eqref{P3}. By \eqref{psmallcond}, \eqref{Jbarbound}, and \eqref{defnorm}, \begin{align} \lvert \alpha'_k \rvert & \le p \varepsilon \frac{1}{8^k} \left( \frac{1}{ 1 - 2 \sum i \lvert \alpha_i \rvert } \right)^{2k} \left( 1 + 2 \sum i \lvert \alpha_i \rvert / p \right)^k \\ & \le p \varepsilon \frac{1}{2^k} \left( \frac{1}{\left( 1 - 2 \| \alpha \| \right)^2} \cdot \frac{ \left( 1 + 2 \| \alpha \| / p \right) }{2} \right)^k \frac{1}{2^k} \end{align} and since $\alpha \in \mathcal{S}$ \begin{align} \le p \varepsilon \frac{1}{2^k} \left( \frac{1}{\left( 1 - 2 \varepsilon p \right)^2} \frac{ \left( 1 + 2 \varepsilon \right)}{2} \right)^k \frac{1}{2^k} \end{align} using \eqref{epssmallcond} \begin{align} \le p \varepsilon \frac{1}{2^k} \frac{1}{2^k}. \end{align} Or \begin{align} 2^k \lvert \alpha'_k \rvert \le p \varepsilon \frac{1}{2^k} \end{align} so that \begin{align} \| \alpha' \| = \sum_2 2^k \lvert \alpha'_k \rvert \le p \varepsilon \sum_2 \frac{1}{2^k} \le p \varepsilon \frac{1}{2} \end{align} and thus $\alpha' \in \mathcal{S}$ as was to be proved. \begin{theorem} \label{theorem_B0val} There is a value of $B_0$ that ensures \begin{align*} \lvert \bar J_n \rvert \le B_0^n, \quad n = 1,2, \dots \end{align*} for all values of $d$. \end{theorem} \begin{theorem} \label{theorem_p0val} There is a value $p_0$ (independent of $d$) such that for $0 \le p < p_0$ the series for $\lambda_d (p)$ in \eqref{ArrangePowerSeries} converges. \end{theorem} Theorem \ref{theorem_p0val} follows from Theorem \ref{theorem_B0val} by the development above. We turn to Theorem \ref{theorem_B0val}. In fact we will see $B_0 = 8e$ works. We could follow the general cluster expansion formalism as given in \cite{BattleFed} and \cite{Brydges}. However in this case it is more elementary to work from the ideas in \cite{BrydgesFed}, and especially the appendix to \cite{BrydgesFed}, due to David Brydges. Now we require the reader to have some familiarity both with \cite{BrydgesFed} and either \cite{FedComp} or Section 5 of \cite{FedFried}. Fortunately these are all rather short. We consider an elegant generalization of the setup in \cite{BrydgesFed}. We replace the configuration space of a single particle, $\mathbb{R}^3$, with individual configurations, points $x \in \mathbb{R}^3$, by the space of two element subsets of $\mathbb{Z}^3$, with individual elements $\left\{ i,j \right\}$, subsets of $\mathbb{Z}^3$. The sum over one dimensional configurations, is changed from \begin{align*} \int dx \end{align*} to \begin{align*} \sum_{\left\{ i,j \right\}} v (i,j) \end{align*} where $v$ is as in (5.6) of \cite{FedFried} or (10) of \cite{FedComp}. Thus we are using the $v$'s to weight the points of the new configuration space. Of the potentials in \cite{BrydgesFed} we keep only $V_r$, given in the Appendix of \cite{BrydgesFed}, in eq (A1). It is constructed from $v_{r2}$ a two-body potential as follows \begin{align} v_{r2} \left( \left\{ i,j \right\} , \left\{ k,l \right\} \right) = \left\{ \begin{array}{l} 0 \quad \left\{ i,j \right\} \cap \left\{ k,l \right\} = \varnothing \\ + \infty \quad \mathrm{otherwise} \end{array} \right. . \end{align} Then $u \left( \left\{ i,j \right\} , \left\{ k,l \right\} \right)$ as defined in (A2) of \cite{BrydgesFed} becomes \begin{align} u \left( \left\{ i,j \right\} , \left\{ k,l \right\} \right) = \left\{ \begin{array}{l} 0 \quad \left\{ i,j \right\} \cap \left\{ k,l \right\} = \varnothing \\ -1 \quad \mathrm{otherwise} \end{array} \right. . \end{align} A natural generalization of (6) of \cite{BrydgesFed} is given by \begin{align} \| u \| = \sum_{j \neq i} \left( \sum_{\left\{ k,l \right\}} \lvert v \left( k,l \right) \rvert \ \lvert u \left( \left\{ i,j \right\} , \left\{ k,l \right\} \right) \rvert \right) \end{align} where $i$ is fixed. It is easy to see from the definition of $v (i,j)$ that \begin{align} \| u \| \le 8 \end{align} since \begin{align} \sum_j \lvert v (i,j) \rvert \le 2. \end{align} The generalization of (56) of \cite{BrydgesFed} becomes \begin{align} \label{gen56} \lvert \bar J_n \rvert \le e^n 8^n. \end{align} (It is easy to improve this, replacing the right side of \eqref{gen56} by $2 e^n 4^n$.) For $d=1$ the expansion in \eqref{ArrangePowerSeries} holds for all $0 \le p \le 1$, as was noted at the end of \cite{FedFried}. We may expect this is true for all $d$!? The methods of the current paper do not get near this result. But the result we have encourages research to address this question. For that matter is $\lambda_d (p)$ analytic in both $p$ and $1 / d $ for $\lvert 1 / d \rvert < 1 $, $\lvert p \rvert < 1$? Or on the other hand perhaps the result of this paper is essentially the best one can do! \begin{thebibliography}{} \bibitem{FedComp} P. Federbush, Computation of Terms in the Asymptotic Expansion of Dimer $\lambda_d$ for High Dimensions, Phys. Lett. A, 374 (2009), 131-133. \bibitem{FedFried} P. Federbush and S. Friedland, An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem, arXiv:math-ph/1011.6579. \bibitem{BattleFed} G.A. Battle and P. Federbush, A Note on Cluster Expansions, Tree Graph Identities, Extra $1/N!$ Factors!!! Lett. Math. Phys. 8, 55 (1984). \bibitem{Brydges} D.C. Brydges, A Short Course in Cluster Expansions, in: \emph{Phenomenes Critiques, Systems Aleatoires, Theories de Gauge,} Parts I, II, Les Houches, 1984, North-Holland, Amsterdam, 1986, pp. 129-183. \bibitem{BrydgesFed} D. Brydges and P. Federbush, A New Form of the Mayer Expansion in Classical Statistical Mechanics, J. Math. Phys. 19, (1978), 2064-2067. \end{thebibliography} \end{document} ---------------1101241108965--