### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu
**Bryan Ek**, Rutgers University, bryan [dot] t [dot] ek {at} math [dot] rutgers [dot] edu

Title: Power series expansion of the free energy for monomer-dimer tilings.

Speaker: **Gleb Pogudin**, New York University

Date: Thursday, February 1, 2018 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

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.

See: http://sites.math.rutgers.edu/~bte14/expmath/