Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

We introduce a family of real random variables (β θ) arising from the supersymmetric nonlinear sigma model H²|² and containing the family β introduced by Sabot, Tarrès, and Zeng (Sabot et al., 2017) in the context of the vertexreinforced jump process. Using this family we construct an exponential martingale generalizing the ones considered in Sabot and Zeng (2018+) and Disertori et al. (2017). Moreover, using the full supersymmetric nonlinear sigma model we also construct a generalization of the exponential martingale involving Grassmann variables.