Paul Federbush
In previous papers an asymptotic expansion for the dimer lambda_d of the form lambda_d ~ (1/2)ln(2d) - 1/2 + c_1/d + c_2/d^2 ... was developed. Kernels J_n were a key ingredient in the theory. Herein we prove J_n are of the form J_n = C_r/d^r + C_(r+1)/d^(r+1) + ... + C_(n-1)/d^(n-1) with r > (n/2)-1.
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In the past few years we have derived asymptotic expansions for lambda_d of the dimer problem and lambda_d(p) of the monomer-dimer problem. The many expansions so far computed are collected herein. We shine a light on results in two dimensions inspired by the work of M. E. Fisher. Much of the work reported here was joint with Shmuel Friedland.
The author and Shmuel Friedland recently presented an expression for lambda_d(p) of the monomer-dimer problem involving a power series in p . Herein I present a simple way to derive this expression for lambda_d(p) from the Mayer (or Virial) series of a dimer gas. The derivation is an exercise in basic statistical mechanics. We expect the relationship to be very useful.
The author has recently presented two different expressions for lambda_d(p) of the monomer-dimer problem involving a power series in p, the first jointly with Shmuel Friedland. These two expressions are certainly equal, but this has not yet been proven rigorously. The first is naturally developed from quantities J_i, cluster expansion kernels. The second from the Mayer (or Virial) series of a dimer gas, in particular from the b_i coefficients in the Mayer series. The sets {b_i} and {J_i} can be derived from each other. Given an arbitrary set of values for either the b_i or the J_i, both expres…
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.
In the development of a presumed asymptotic expansion for lambda_d in previous work, a basic step involved extracting the asymptotic behavior of a sum as being dominated by the largest term in the sum. But this argument only is valid when the terms in the sum are positive. In the case terms in the sum are not all positive (some of the J_n are negative) we replace the `largest term' argument by the route of this paper. The sum is replaced by a contour integral. We then find a stationary point of the integrand that we conjecture dominates the asymptotic behavior of the integral.
The dimer problem arose in a thermodynamic study of diatomic molecules, and was abstracted into one of the most basic and natural problems in both statistical mechanics and combinatoric mathematics. Given a rectangular lattice of volume V in d dimensions, the dimer problem loosely speaking is to count the number of different ways dimers (dominoes) may be layed down on the lattice to completely cover it. It is known that the number of such coverings is roughly exp(lambda_d V) for some number lambda_d. The first terms in the expansion of lambda_d have been known for about thirty years lambda_d ~…
Working with a presumed asymptotic series for lambda_d developed in previous work, we make some intelligent guesses for lambda_d with d=3, 4, 5; and estimates for the corresponding errors. We present arguments in favor of these guesses, we earnestly believe they will turn out to be correct. Such approximate values may help stimulate people working on rigorous bounds. In addition to suggesting bounds to prove, there will be the strong motivation to prove me wrong.
In a recent paper S. Friedland and the author presented a formal expression for Lambda_d(p) of the monomer-dimer problem in d dimensions involving a power series in p. We there presented the result of computations for the terms in the power series through the sixth power. I herein present the result for the seventh power term, in d = 2. An interesting feature of the new computation is that it would have been impossible without a new algorithm for computation of the Tutte polynomial, by Bjorklund, Husfeldt, Kaski, and Koivisto.
In a recent paper S. Friedland and the author presented a formal expression for lambda_d(p) of the monomer-dimer problem on a d-dimensional rectangular lattice, which involved a power series in p. Herein, we find simlar expressions for lambda_tri(p) and lambda_hex(p), the entropy per site for triangular and hexagonal lattices.
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).
In a previous paper an asymptotic expansion for lambda_d in powers of 1/d was developed. The results of computer computations for some terms in the expansion, as well as various quantities associated to the expansion, are herein presented. The computations (in integer arithmetic) are actually done only for d=1, d=2, and d=3, but some results for arbitrary dimension follow from the general structure of the theory. In particular we obtain the next term in powers of 1/d, the 1/d^3 term in the asymptotic expansion for lambda_d, as well as checking the correctness of the terms previously calculated…
The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda…
In these lecture notes we present some connections between random matrices, the asymmetric exclusion process, random tilings. These three apparently unrelated objects have (sometimes) a similar mathematical structure, an interlacing structure, and the correlation functions are given in terms of a kernel. In the basic examples, the kernel is expressed in terms of orthogonal polynomials.
Split Two-Periodic Aztec Diamond
本研究通过结合两种不同双周期权重，提出一种打破周期性的Aztec菱形双随机模型，并计算其相关核及局部渐近行为，揭示了其与典型双周期模型在最高阶一致但次高阶项不同的特性。
We work with the following expression for the entropy (density) of a dimer gas on an infinite r-regular lattice lambda(p) = 1/2 [ pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sum converges for density, p, small enough. Pernici has computed the coefficients d_k for k < 13. He found these d_k to be polynomials in certain interesting "geometric quantites" arising in the work of Wanless. Each of these quantities is the number density of isomorphic mappings of some graph into the lattice (graph). So for a bipartite lattice d_2 = c_2 d_3 = c_3 d_4 = c_4 + c_5 hat{G}_1 d_5 =…
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