PC Seminar: Boundary correlations in planar LERW and UST revisited

Abstract: We consider one of the simplest and most studied model of a random tree: the uniform spanning tree (UST). It is intimately related to simple random walk: Wilson's algorithm shows that a sample from UST can be generated by running iteratively simple random walks with their loops removed (termed loop-erased random walks, LERW). In planar domains, this is a critical model in the sense that it is expected that its scaling limit is described by a conformal field theory (CFT). Indeed, we provide a CFT description of boundary effects in this model, which has central charge $c=-2$ (and in particular, the full model should be a logarithmic CFT). Using the relation with LERWs, we can find explicit determinantal, combinatorial formulas for probabilities of UST boundary connection events. From this, we can prove their convergence to certain CFT quantities (which we call conformal blocks) in the scaling limit. To verify the expected CFT properties, we combine probabilistic techniques with representation theory.

This is a seminar in our seminar series on Probability and Combinatorics (PC).