Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

TL;DR

This paper introduces a new family of random variables from the supersymmetric nonlinear sigma model, generalizing existing martingales and incorporating Grassmann variables, advancing the mathematical understanding of these models.

Contribution

It constructs a generalized exponential martingale from the supersymmetric sigma model, extending previous work and including Grassmann variables for broader applicability.

Findings

New family of random variables from supersymmetric models

Generalized exponential martingale including Grassmann variables

Connections to vertex-reinforced jump processes

Abstract

We introduce a family of real random variables $(\beta,\theta)$ arising from the supersymmetric nonlinear sigma model and containing the family $\beta$ introduced by Sabot, Tarr\`es, and Zeng [STZ17] in the context of the vertex-reinforced jump process. Using this family we construct an exponential martingale generalizing the one considered in [DMR17]. Moreover, using the full supersymmetric nonlinear sigma model we also construct a generalization of the exponential martingale involving Grassmann variables.