Schramm's formula for multiple loop-erased random walks

TL;DR

This paper revisits and extends Schramm's formula for loop-erased random walks, providing explicit results in various geometries and generalizing to multiple walks, using complex connections and Green functions.

Contribution

It introduces a new explicit formula for multiple loop-erased random walks in different geometries, extending previous single-walk results.

Findings

Explicit formulas in the scaling limit for various geometries

Extension of Schramm's formula to multiple walks

Use of complex connections and Green functions

Abstract

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.