Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems

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.Comment: 4 page

A phase cell approach to Yang-Mills theory V. Analysis of a chunk

In the present formalism the Yang-Mills field is constructed as a “non-linear sum” of excitations, small field excitations, the modes, and large field excitations, the chunks. The chunk excitations, herein studied, are each described by a finite number of group element variables. The continuum field associated to the excitation in general has point gauge singularities (arising from the non-trivial π 3 (G)). We find estimates for plaquette assignments, edge assignments, and the smoothness of edge assignments, at all scales. The central conceptual motor in our constructions and estimates is a split up of the field at each length scale, locally, into a pure gauge field, and a deviation field. An example is presented establishing the general inevitability of gauge singularities, as a consequence of fall off requirements on the continuum field of an excitation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46472/1/220_2005_Article_BF02104497.pd

A phase cell approach to Yang-Mills theory

Variables are chosen to describe the continuum Yang-Mills fields, a discrete set of group valued variables. These are group elements associated to the sequence of lattice field theory configurations realizing the continuum field. The field is “laid down” inductively. At each inductive step one of three types of “field excitations” makes its contribution to the total field. These are either “pure modes”, “averaging correction modes”, or “chunks”. The pure modes are small field excitations, as studied in previous papers in this series [2,3]. The averaging correction modes are small excitations added to make sure the block spin transformation is satisfied at each edge. The chunks, encompassing most of our difficulties, are large field excitations. Topological obstructions in π 3 ( G ) must be dealt with in defining a gauge choice for each chunk. The laying down process is complex, but fiendishly clever, ensuring a principle of “gauge invariant coupling”. Each group valued variable is either the “amplitude” of a pure mode or an “internal variable” in a chunk. The amplitude of an averaging correction mode is a dependent variable, a function of the (independent) variables used to describe the field. The (independent) variables herein defined are those whose mutual interaction will later be inductively decoupled in defining the phase cell cluster expansion (of course treating the variables of each chunk as a unit).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46469/1/220_2005_Article_BF01225039.pd

A phase cell approach to Yang-Mills theory

In this paper the basic local stability result is obtained, in a form valid in both small field and large field regions. To achieve this, some modifications are made in both the action and the renormalization group transformation. Though there is some sacrifice of elegance in these modifications, the establishment of this local stability estimate yields the most basic ingredient of the phase cell cluster expansion, good estimates for all the actions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46468/1/220_2005_Article_BF01207369.pd

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

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).Comment: 15 pages, much more about d=1,2,