Free boundary dimers: random walk representation and scaling limit

Wei Qian (1, 2) , Nathanaël Berestycki (3) , Marcin Lis (3)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 CNRS - Centre National de la Recherche Scientifique
3 University of Vienna [Vienna]

We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight z > 0 to the total weight of the configuration. A bijection described by Giuliani, Jauslin and Lieb [15] relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of z > 0, the scaling limit of the height function is the Gaussian free field with Neumann (or free) boundary conditions, thereby answering a question of Giuliani et al.

Domaines: Mathématiques [math]; Probabilités [math.PR]; Physique mathématique [math-ph]; Variables complexes [math.CV]

Wei Qian, Nathanaël Berestycki, Marcin Lis. Free boundary dimers: random walk representation and scaling limit. 2021. ⟨hal-03370297⟩