UST branches, martingales, and multiple SLE(2)

Abstract

We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple $\mathrm {SLE}(2)$, i.e., an $\mathrm {SLE}(2)$ process weighted by a suitable partition function. By recent results, this also characterizes the “global” scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with $N$ branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence proof for a boundary-visiting UST branch and a boundary-visiting $\mathrm {SLE}(2)$.