Self-avoiding walks on hyperbolic graphs

We study self-avoiding walks (SAWs) on non-Euclidean lattices that correspond to regular tilings of the hyperbolic plane ('hyperbolic graphs'). We prove that on all but at most eight such graphs, (i) there are exponentially fewer N-step self-avoiding polygons than there are N-step SAWs, (ii) the number of N-step SAWs grows as μ_w^N within a constant factor, and (iii) the average end-to-end distance of an N-step SAW is approximately proportional to N. In terms of critical exponents from statistical physics, (ii) says that γ = 1 and (iii) says that v = 1. We also prove that γ is finite on all hyperbolic graphs, and we prove a general identity about non-reversing walks that had previously been discovered for certain special cases.