# Boundary correlations in planar LERW and UST

## TL;DR

This paper derives explicit formulas for boundary visit probabilities in planar LERW and UST, demonstrating their convergence to conformally covariant functions satisfying PDEs, linking to SLE and conformal field theory.

## Contribution

It provides the first explicit formulas for boundary event probabilities in planar LERW and UST, connecting these to conformal invariance and SLE partition functions.

## Key findings

- Probabilities converge to conformally covariant functions
- Derived PDEs of second and third order for these functions
- Formulas for SLE$_2$ partition functions at criticality

## Abstract

We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. 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. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple $\mathrm{SLE}_\kappa$ at $\kappa=2$.