Abstract:
The dimer model on a finite bipartite graph is a uniformly chosen perfect matching. It is a classical model of mathematical physics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s.
A central object for the dimer model is a notion of height function introduced by Thurston, which turns the dimer model into a random discrete surface. I will discuss a series of recent results with Benoit Laslier (Paris) and Gourab Ray (Victoria) where we establish the convergence of the height function to the Gaussian free field in a variety of situations. Unlike previous works on the dimer model, we do not exploit the integrable (determinantal) structure of the model but instead rely on couplings between the Gaussian free field and SLE curves known as Imaginary Geometry. As a consequence, our result show a strong form of universality. Time permitting, I will discuss (the beginning of) a programme where this result is extended to the setting of Riemann surfaces.
Based on joint work with Benoit Laslier (Paris) and Gourab Ray (Victoria).