Double dimers on planar hyperbolic graphs via circle packings

TL;DR

This paper investigates the double dimer model on hyperbolic graphs using circle packings, establishing conditions for weak limits, and proving the absence of bi-infinite paths and percolation properties of the height function.

Contribution

It introduces a new approach to analyze double dimer models on hyperbolic graphs via circle packings and proves key properties of the model's behavior.

Findings

Weak limit of the dimer model exists under specific boundary convergence conditions.

Double dimer model has no bi-infinite path almost surely.

Height function exhibits double exponential tail and does not percolate for large heights.

Abstract

In this article we study the double dimer model on hyperbolic Temperleyan 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 the 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, we prove that the double dimer model has no bi-infinite path almost surely. Along the way we prove that 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. 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.