arXiv subjects
Publications and source records attributed to Paul Federbush.
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.
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.
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.
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).
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.
We consider tiles (dimers) each of which covers two vertices of a rectangular lattice. There is a normalized translation invariant weighting on the shape of the tiles. We study the pressure, p, or entropy, (one over the volume times the logarithm of the partition function). We let p_0 (easy to compute) be the pressure in the limit of absolute smoothness (the weighting function is constant). We prove that as the smoothness of the weighting function, suitably defined, increases, p converges to p_0, uniformly in the volume. It is the uniformity statement that makes the result non-trivial. In an earlier paper the author proved this, but with an additional requirement of a certain fall-off on the weighting function. Herein fall-off is not demanded, but there is the technical requirement that each dimer connect a black vertex with a white vertex, vertices colored as on a checker board. This seems like a very basic result in the theory of pressure (entropy) of tilings.
In recent work the author presented a formal expansion for lambda_d associated to the dimer problem on a d-dimensional rectangular lattice. Expressed in terms of d it yielded a presumed asymptotic expansion for lambda_d in inverse powers of d. We also considered an expansion in powers of x, a formal variable ultimately set equal to 1. We believe this series has better asymptotic properties than the expansion in inverse powers of d. We discuss this, and apply the same method to the two-dimensional triangular and hexagonal lattices. Viewed as a test of the x-expansion the results on those two lattices are satisfactory, if not thoroughly convincing.
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 laid down in the lattice (without overlapping) to completely cover it. Each dimer covers two neighboring vertices. It is known that the number of such coverings is roughly exp(lambda_d V) for some constant lambda_d as V goes to infinity. Herein we present a mathematical argument for an asymptotic expansion for lambda_d in inverse powers of d, and the results of computer computations for the first few terms in the series. As a glaring challenge, we conjecture no one will compute the next term in the series, due to the requisite computer time and storage demands.
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.
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.
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 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 by hand. For d=2 and d=3 a second expansion for lambda_d is defined and numerically studied. Contrary to our hopes this second type of expansion appears not to be convergent. There is not enough evidence to argue if the expansions for lambda_d in powers of 1/d are convergent for either d=2 or d=3, but likely they are not. (Certainly convergence, but at a very slow rate, remains an interesting possibility in all our cases.)
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 ~ (1/2)ln(2d)-1/2 Herein we present a mathematical argument for an asymptotic expansion lambda_d ~ (1/2)ln(2d) -1/2 +(1/8)/d + (5/96)/d^2 +... with the first few terms given explicitly.
We consider tiles of some fixed size, with an associated weighting on the shapes of tile, of total mass 1. We study the pressure, $p$, of tilings with those tiles; the pressure, one over the volume times the logarithm of the partition function. (The quantity we define as "pressure" could, perhaps equally harmoniously with physics notation, be called "entropy per volume", neither nomenclature is "correct".) We let $\hat p^0$ (easy to compute) be the pressure in the limit of absolute smoothness (the weighting function is constant). Then as smoothness, suitably defined, increases, $p$ converges to $\hat p^0$, uniformly in the volume. It is the uniformity requirement that makes the result non-trivial. This seems like a very basic result in the theory of pressure of tilings. Though at the same time, perhaps non-glamorous, being bereft of geometry and not very difficult. The problem arose for us out of study of a problem in mathematical physics, associated to a model of ferromagnetism.
We consider the group of permutations of the vertices of a lattice. A random walk is generated by unit steps that each interchange two nearest neighbor vertices of the lattice. We study the heat equation on the permutation group, using the Laplacian associated to the random walk. At t=0 we take as initial conditions a probability distribution concentrated at the identity. A natural conjecture for the probability distribution at long times is that it is 'approximately' a product of Gaussian distributions for each vertex. That is, each vertex diffuses independently of the others. We obtain some formal asymptotic results in this direction. The problem arises in certain ways of treating the Heisenberg model of ferromagnetism in statistical mechanics.
A criterion for ferromagnetism is presented suggesting a line of proof to rigorously establish the phase transition. Spectral information will be required in certain invariant subspaces of the Hamiltonian, hopefully the relatively crude estimates needed will be not too difficult to establish.
Certain approximations for the spin expectations have been studied both numerically and theoretically. Approximations are constructed using solutions of the lattice heat equation. The most interesting idea is a certain randomness assumption for a theoretically derived quantity, that leads directly to a given approximation. We are betting the randomness assumption is rigorously correct.
A polymer expansion is given for the Quantum Heisenberg Ferromagnet wave function. Working on a finite lattice, one is dealing entirely with algebraic identities; there is no question of convergence. The conjecture to be pursued in further work is that effects of large polymers are small. This is relevant to the question of the utility of the expansion and its possible extension to the infinite volume. In themselves the constructions of the present paper are neat and elegant and have surprising simplicity.