Marianna Russikikh , Playing dominos in different domains

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.