Large Structures Seminar: Kalle Kytölä

The goal of the talk is to present a few related exact scaling limit results for correlations of two related stochastic models on planar graphs. The loop erased random walk is constructed by taking a suitable random walk on a graph, and chronologically erasing loops from its trajectory to obtain a simple path on the graph. We consider loop-erased random walks on graph approximations of a planar Jordan domain, and consider the probabilities with which these walks use given N edges near the boundary, in a given order. For regular enough graph approximations of the domain, we show that the probability is proportional to $a^{3N}$, where $a$ is the mesh-size of the graph. We also show that as the mesh size tends to zero, the limit of the probability divided by $a^{3N}$ exists and is a non-trivial conformally covariant quantity. Importantly, the limit solves a system of two second order and N third order linear partial differential equations, as had been predicted using conformal field theory. Similar results are presented also for the uniformly random spanning tree of the graph. The talk is based on joint work with Alex Karrila and Eveliina Peltola.