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

TL;DR

This paper analyzes a supersymmetric sigma model, linking it to random spanning forests, and shows how pinning affects correlation decay, with implications for understanding complex network behaviors.

Contribution

It establishes a probabilistic interpretation of the two-point correlation function and compares effects of general pinning versus pinning at one point.

Findings

Correlation function relates to connectivity in rooted forests.

Pinning at one point dominates general pinning effects.

Exponential decay of correlations observed in ladder graphs.

Abstract

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.