Double dimers on planar hyperbolic graphs, via circle packings

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.

Gourab Ray received the Bachelor of Statistics and Master of Statistics degrees from the Indian Statistical Institute, Kolkata, in 2008 and 2010, respectively, and the Ph.D. degree in Mathematics from University of British Columbia in 2014. He was a Research Associate with DPMMS at Cambridge University from 2014 to 2017. He was an Assistant Professor at University of Victoria fro 2017 to 2021. He is currently an Associate Professor at the Department of Mathematics and Statistics at University of Victoria. His research interests include Statistical mechanics and phase transitions, Coding of statistical mechanics models and structured signal detection problems in high dimensional and network models.