Mathematical Physics Seminar

We consider a simple random walk on the two-dimensional torus (Z/NZ)^2, started from the uniform distribution, as N becomes large. We investigate the local limit of this walk, when run up to a suitable time while forced to avoid a given point; that is, we look at the trace this walk leaves in a sufficiently small region on the torus. As will be explained in detail, by "poissonizing" the excursions of this walk, one is naturally led to consider a Poisson point process comprising (pieces of) random walk trajectories avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning their intensity with N, the occupation field of these pieces can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Among other things, this construction allows to link the local times of the random walk to the pinned massless field in Z^2 by means of a suitable isomorphism theorem, reminiscent of representation formulas due to Symanzik, Brydges-Froehlich-Spencer, Dynkin et al.