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

2020 
We study the behaviour of a sequence of biased random walks $$(X^{\scriptscriptstyle (i)})_{i \ge 0}$$ on a sequence of random graphs, where the initial graph is $$\mathbb {Z}^d$$ 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 $$(X^{\scriptscriptstyle (i)})_{i \ge 1}$$ is (transient but) not ballistic, and the limiting graph is an infinite simple (self-avoiding) path. We also give examples for which each $$(X^{\scriptscriptstyle (i)})_{i \ge 1}$$ is ballistic, but the limiting graph is not a simple path.
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