Biased Random Walk on the Trace of Biased Random Walk on the Trace of …

Abstract

We study the behaviour of a sequence of biased random walks (X(i))i≥0 on a sequence of random graphs, where the initial graph is Zd and otherwise the graph for the ith walk is the trace of the (i−1)st walk. The sequence of bias vectors is chosen so that each walk is transient. We prove the aforementioned transience and a law of large numbers, and provide criteria for ballisticity and sub-ballisticity. We give examples of sequences of biases for which each (X(i))i≥1 is (transient but) not ballistic, and the limiting graph is an infinite simple (self-avoiding) path. We also give examples for which each (X(i))i≥1 is ballistic, but the limiting graph is not a simple path.