LA Probability Forum

UCLA, Math Sciences Room 6627

A programme initiated by Makarov and Smirnov is to describe near-critical scaling limits of planar statistical mechanics models in terms of massive SLE and/or Gaussian free field. We will consider in this talk the dimer model on the square and hexagonal lattices with doubly periodic weights. Although in the near-critical regime the Kasteleyn matrix is related to a massive Laplacian, Chhita proved that on the whole plane square lattice, the corresponding height function has a scaling limit which surprisingly is not even Gaussian.

In joint work with Levi Haunschmid (TU Vienna) we obtain some new results about this model in three different directions: (a) we establish a rigorous connection with the massive SLE$_2$ constructed by Makarov and Smirnov; (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. Our techniques rely on Temperley's bijection and the "imaginary geometry" approach developed in earlier work with Benoit Laslier and Gourab Ray.