ar X iv : m at h / 06 11 77 5 v 1 [ m at h . PR ] 2 5 N ov 2 00 6 Colored loop - erased random walk on the complete graph ∗

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 coloring of these edges by painting edges with a fixed color as long as the walk does not loop back on itself, then switching to a new color whenever a loop is erased, with each new color distinct from all previous colors. The pattern of colors 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. denote the event that the loop-erased walk from N to [k] has i + 1 or more edges, and the i th and (i + 1) th of these edges are colored 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 colored segments of the loop-erased walk, and yields asymptotic descriptions of these random lengths as N → ∞.