Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

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.