Biased random walk on the trace of biased random walk on the trace of.. [article]

David Croydon, Mark Holmes
2019 arXiv   pre-print
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 i-th 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
more » ... allistic, 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.
arXiv:1901.04673v1 fatcat:hlqlupn2drdafjuslk2dgyxt3u