Berestycki, N., Lis, M., & Qian, W. (2023). Free boundary dimers: random walk representation and scaling limit. Probability Theory and Related Fields, 186(3–4), 735–812. https://doi.org/10.1007/s00440-023-01203-x
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 et al. (J Stat Phys 163(2):211-238, 2016) 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 centered height function is the Gaussian free field with Neumann (or free) boundary conditions. It is the first example of a discrete model where such boundary conditions arise in the continuum scaling limit.
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Project title:
Spins, Felder und Schleifen: 59944 / P 36298 (FWF - Österr. Wissenschaftsfonds)