Dominos in hedgehog domains

TL;DR

This paper introduces 'hedgehog domains' as a new class of discrete approximations for planar regions, demonstrating that dimer model height fluctuations converge to the Gaussian Free Field and revealing connections to fermionic observables and harmonicity in the scaling limit.

Contribution

It defines hedgehog domains and proves convergence of dimer height fluctuations to the Gaussian Free Field, linking to fermionic boundary conditions and harmonicity in the double-dimer model.

Findings

Height fluctuations converge to Gaussian Free Field with Dirichlet boundary conditions.

Dimer coupling functions satisfy Riemann-type boundary conditions similar to the Ising model.

Double-dimer height function's expectation becomes harmonic in the scaling limit.

Abstract

We introduce a new class of discrete approximations of planar domains that we call "hedgehog domains". In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height function of a random dimer tiling on hedgehog discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. Interestingly enough, in this case the dimer model coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model. In addition, using the same factorization of the double-dimer model coupling function as in [17], we show that in the case of approximations by hedgehog domains the expectation of the double-dimer height function is harmonic in the scaling limit.