Near-critical dimers and massive SLE

TL;DR

This paper establishes a rigorous connection between the near-critical dimer model on lattices and the massive SLE$_2$ process, demonstrating universality and conformal covariance of the scaling limit in various domains.

Contribution

It introduces an inhomogeneous near-critical dimer model and proves its convergence to massive SLE$_2$, extending previous results to arbitrary bounded domains with Temperleyan boundary conditions.

Findings

Proves convergence of the dimer model to massive SLE$_2$ in bounded domains.

Shows universality and conformal covariance of the scaling limit.

Introduces an exact discrete Girsanov identity on the triangular lattice.

Abstract

We consider the dimer model on the square and hexagonal lattices with doubly periodic weights. The purpose of this paper is threefold: (a) we establish a rigourous connection with the massive SLE$_2$ constructed by Makarov and Smirnov (and recently revisited by Chelkak and Wan); (b) we show that the convergence takes place in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. For this we introduce an inhomogeneous near-critical dimer model, corresponding to a drift for the underlying random walk which is a smoothly varying vector field or alternatively to an inhomogeneous mass profile. When the vector field derives from a log-convex potential we prove that the corresponding loop-erased random walk has a universal scaling limit. Our techniques rely on an exact discrete Girsanov identity on the triangular lattice which may be of independent interest. We complement our results by stating precise conjectures making connections to a generalised Sine-Gordon model at the free fermion point.