Gleb Pogudin, New York University (NYU)

Abstract

Let a(n, p), where 0 < p < 1, be the number of tilings of n-by-n square with dimers and monomers, such that dimers occupy approximately p of the area. Then the following limit characterizes the growth of this number and is called the free energy of the monomer-dimer system:

f(p) = lim log( a(n, p) ) / n^2 with n -> infinity

It is an open problem how to compute f(p). I will describe a method allowing to compute the first 63 terms of its Taylor expansion at zero. This yields to accurate estimates for many values of p. I will also discuss other approaches to computing f(p) and interesting mathematical problems connected with them.