Perfect Matchings and Essential Spanning Forests in Hyperbolic Double Circle Packings

We investigate perfect matchings and essential spanning forests in planar hyperbolic graphs. Using the double circle packing for a pair of dual graphs, we relate the inverse of the weighted adjacency matrix to the difference of Green's functions plus an explicit harmonic Dirichlet function. We prove that the infinite-volume Gibbs measure obtained from approximations by finite domains with exactly two convex white corners converging to two distinct points along the boundary is extremal, yet not invariant with respect to a finite-orbit subgroup of the automorphism group. We then prove that the variance of the height difference of two i. i. d. ~uniformly weighted perfect matchings under the boundary condition above, or the Temperley boundary condition on a transitive nonamenable planar graph is always finite; in contrast to the 2D uniformly weighted dimer model on a transitive amenable planar graph with Temperley boundary conditions as proved in RK01, KOS06, where the variance of height difference grows in the order of n, with n being the graph distance to the boundary. This also implies that a. s. ~each point is surronded by finitely many cycles in the symmetric difference of two i. i. d. ~perfect matchings, also in contrast to the 2D Euclidean case.