Title: “Boundary correlations in planar LERW and UST.”

Abstract: We present explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks (LERW), as well as for connectivity events for branches in the uniform spanning tree (UST). We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory (CFT). This is among the first rigorously established scaling limit results showing CFT predicted PDEs of order higher than two. Apart from combinatorics specific to LERWs and USTs, the proof uses also a more conceptual reasoning: a fusion argument together with an interchange of the scaling limit with a short distance expansion of the correlation functions.
The talk is based on joint work with Alex Karrila and Eveliina Peltola.