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

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 $\infty $. In this note we derive expressions for transition probabilities for this model in dimensions $d \ge 2$. For $d=2$ 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 $z \mapsto \sqrt{z} $.