We study the behaviour of a sequence of biased random walks (X(i))i≥0\documentclass[12pt]{minimal}
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\begin{document}$$(X^{\scriptscriptstyle (i)})_{i \ge 0}$$\end{document} on a sequence of random graphs, where the initial graph is Zd\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^d$$\end{document} and otherwise the graph for the ith walk is the trace of the (i-1)\documentclass[12pt]{minimal}
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\begin{document}$$(i-1)$$\end{document}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\documentclass[12pt]{minimal}
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\begin{document}$$(X^{\scriptscriptstyle (i)})_{i \ge 1}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$(X^{\scriptscriptstyle (i)})_{i \ge 1}$$\end{document} is ballistic, but the limiting graph is not a simple path.
