Abstract
The loop-erased random walk (LERW) in $ {\mathbb {Z}}^{d}, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.
Highlights
In this paper we establish the existence of the infinite two-sided loop-erased random walk (LERW)
The LERW is the measure on non self-intersecting paths obtained by erasing loops chronologically from a simple random walk
We will give a definition that is valid for d ≥ 2 that is seen to be equivalent to the usual definition for d ≥ 3
Summary
In this paper we establish the existence of the infinite two-sided loop-erased random walk (LERW). The (infinite, one-sided) LERW is the measure on non self-intersecting paths obtained by erasing loops chronologically from a simple random walk. For d ≥ 3, one gets the same measure by taking a simple random walk without conditioning and defining σ0 = max{j : Sj = 0} This is the original definition as in [5], but it is often useful to view this probability measure on infinite self-avoiding paths as a. If we let pk be p restricted to Ak, {pk} is a consistent family of probability measures and induces a probability measure on pairs of infinite self-avoiding paths starting at the origin that do not intersect (other than the initial point) We call this process the two-sided infinite loop-erased random walk. We view this measure as giving transition probabilities for LERW stopped when it reaches a smaller radius n and this is the process that we couple
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