# Student Probability Seminar

#### Planar Dimer Model and Conformal Invariance

Speaker: Titus Lupu

Location: Warren Weaver Hall 202

Date: Friday, May 6, 2011, 3:50 p.m.

Synopsis:

The dimer model is a statistical physics model studied in sixties by Pieter Kasteleyn and which has experienced important advances in the last decade. One studies random uniformly sampled domino tilings of a polyomino. To such a tilling corresponds a "height function" which is an integer valued function on the squares of the polyomino. When a simply connected domain D in the complex plane gets approximated in an appropriate way by a sequence of polyominos with increasingly larger number of increasingly smaller squares, it was conjectured that the fluctuation of the corresponding random height functions away from their mean values converges to the Gaussian Free Field on D with Dirichlet boundary conditions. This result was eventually proved in 2008 by Richard Kenyon. He also computed the limit of the mean value of height functions. I my talk I will present some basic combinatorial results on random domino tilings and expound the main ideas of Kenyon's proof.