A comparison of a nonlinear sigma model with general pinning and pinning at one point

We study the nonlinear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991. This model can be related to the mixing measure of a vertex-reinforced jump process. We prove that the two-point correlation function has a probabilistic interpretation in terms of connectivity in rooted random spanning forests. Using this interpretation, we dominate the two-point correlation function for general pinning, e.g. for uniform pinning, with the corresponding correlation function with pinning at one point. The result holds for a general finite graph, asymptotically as the strength of the pinning converges to zero. Specializing this to general ladder graphs, we deduce in the same asymptotic regime exponential decay of correlations for general pinning.