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Merkl, Franz; Rolles, Silke W. W.; Tarres, Pierre (2019): Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model. In: Probability Theory and Related Fields, Vol. 173, No. 3-4: pp. 1349-1387
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Abstract

In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarres of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees.