  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.
