Coloured Loop-Erased Random Walk on the Complete Graph

Abstract

Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from i to j if the loop-erased walk makes a step from i to j. We introduce a colouring of these edges by painting edges with a fixed colour as long as the walk does not loop back on itself, then switching to a new colour whenever a loop is erased, with each new colour distinct from all previous colours. The pattern of colours along the edges of the loop-erased walk then displays stretches of consecutive steps of the walk left untouched by the loop-erasure process. Assuming that the underlying sequence generating the loop-erased walk is a sequence of independent random variables, each uniform on [N] := {1, 2, . . ., N}, we condition the walk to start at N and stop the walk when it first reaches the subset [k], for some 1 ≤ k ≤ N − 1. We relate the distribution of the random length of this loop-erased walk to the distribution of the length of the first loop of the walk, via Cayley's enumerations of trees, and via Wilson's algorithm. For fixed N and k, and i = 1, 2, . . ., let Bi denote the events that the loop-erased walk from N to [k] has i + 1 or more edges, and the ith and (i + 1)th of these edges are coloured differently. We show that, given that the loop-erased random walk has j edges for some 1 ≤ j ≤ N − k, the events Bi for 1 ≤ i ≤ j − 1 are independent, with the probability of Bi equal to 1/(k + i + 1). This determines the distribution of the sequence of random lengths of differently coloured segments of the loop-erased walk, and yields asymptotic descriptions of these random lengths as N → ∞.