UST branches, martingales, and multiple SLE(2) [article]

Alex Karrila
2020 arXiv   pre-print
We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple SLE(2), i.e., an 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
more » ... unctions 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 SLE(2).
arXiv:2002.07103v2 fatcat:6t7c3i7jsnhvpawstb2a6s5w4i