A supersymmetric approach to martingales related to the vertex-reinforced jump process

TL;DR

This paper presents a supersymmetric framework to derive an infinite hierarchy of martingales related to the vertex-reinforced jump process, providing an alternative to previous methods and unveiling new mathematical structures.

Contribution

It introduces a supersymmetric approach to derive an infinite hierarchy of martingales associated with the vertex-reinforced jump process, expanding the theoretical understanding.

Findings

Derived two key martingales from a supersymmetric hyperbolic sigma model.

Established an infinite hierarchy of martingales generated by a new function.

Provided an alternative derivation method for martingales related to vertex-reinforced processes.

Abstract

Sabot and Zeng have discovered two martingales, one of which played a key role in their investigation of the vertex-reinforced jump process. Starting from the related supersymmetric hyperbolic sigma model, we give an alternative derivation of these two martingales. They turn out to be the first two instances in an infinite hierarchy of martingales, derived from a generating function.