Title: Playing dominos in different domains
Abstract:
We will discuss an extension of several results of Richard Kenyon on
the dimer model. In 1999 he has shown that fluctuations of the height
function of a random dimer tiling on Temperley discretizations of a
planar domain converge in the scaling limit to the Gaussian Free Field
with Dirichlet boundary conditions. We will discuss an extension of
this result to other classes of discretizations. In particular, we
will discuss boundary conditions of the coupling function in the
so-called "even" domains. Interestingly enough, in this case, the
coupling function satisfies the same Riemann-type boundary conditions
as fermionic observables in the Ising model. The main tool is a
factorization of the gradient of the expectation of the height
function in the double-dimer model into a product of two discrete
holomorphic functions. In particular, we use this factorization to
show that, rather surprisingly, the expectation of the double-dimer
height function in the Temperley case is exactly discrete harmonic
even before taking the scaling limit.