Transition probabilities for infinite two-sided loop-erased random walks

TL;DR

This paper derives explicit transition probabilities for the infinite two-sided loop-erased random walk in dimensions two and higher, connecting these probabilities to discrete harmonic functions and Laplacian operators.

Contribution

It provides new formulas for transition probabilities of the infinite two-sided LERW, linking them to discrete harmonic functions and determinants, especially in the plane.

Findings

Explicit formulas for transition probabilities in 2D and higher dimensions.

Connection between transition probabilities and discrete harmonic functions.

Representation of formulas using Laplacian with signed weights and determinants.

Abstract

The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the `middle part' of an infinite LERW loop going through 0 and infinity. In this note we derive expressions for transition probabilities for this model in dimensions two and up. In the plane, the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of the complex square-root.