Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

TL;DR

This paper demonstrates that a certain reinforced stochastic process converges to an extended supersymmetric hyperbolic nonlinear sigma model, providing insights into its asymptotic behavior and related quantities.

Contribution

It introduces an extension of Zirnbauer's supersymmetric hyperbolic nonlinear sigma model and proves its emergence as a limit of a time-changed vertex-reinforced jump process.

Findings

Convergence of the process to the sigma model.

Description of asymptotics for crossing numbers and local times.

Analysis of path endpoints and last exit trees.

Abstract

In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarr\`es of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees.