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

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model