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Sabot and Zeng have discovered two martingales, one of which played a keyrole in their investigation of the vertex-reinforced jump process. Startingfrom the related supersymmetric hyperbolic sigma model, we give an alternativederivation of these two martingales. They turn out to be the first twoinstances in an infinite hierarchy of martingales, derived from a generatingfunction.

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