UBC Probability Seminar: Gourab Ray

In this article we study the double dimer model on hyperbolic Temper- leyan graphs via circle packings. We prove that on such graphs, the weak limit of the dimer model exists if and only if the removed black vertex from the boundary of an exhaustion converges to a point on the unit circle in the circle packing representation of the graph. One of our main results is that for such measures, the double dimer model has no bi-infinite path almost surely. Along the way we prove that in the nonamenable setup, the height function of the dimer model has double exponential tail and faces of height larger than k do not percolate for large enough k. All of these results are new, even for hyperbolic lattices. The proof uses the connection between winding of uniform spanning trees and dimer heights, the notion of stationary random graphs, and the boundary theory of random walk on circle packings.