Hyperbolic symmetry and random walk isomorphisms
The classical random walk isomorphism theorems relate the square of a Gaussian free field to the
local time of a corresponding random walk. These relations and their supersymmetric versions
have been used in both directions: in the study of (non-Gaussian) spin systems and field
theories using random walk techniques and in the study of self-interacting walks in terms of
fields. I will present non-Gaussian analogues that relate hyperbolic sigma models to linearly
reinforced random walks. As an application, we show that the vertex-reinforced jump process is
recurrent in two dimensions.
This is joint work with Andrew Swan and Tyler Helmuth.