[hal-00962241] The potential energy of biased random walks on trees

Biased random walks on supercritical Galton--Watson trees are introduced and studied in depth by Lyons 1990 and Lyons Pemantle and Peres 1996 We investigate the slow regime in which case the walks are known to possess an exotic maximal displacement of order $\log n^3$ in the first $n$ steps Our main result is another --- and in some sense even more --- exotic property of biased walks the maximal potential energy of the biased walks is of order $\log n^2$ More precisely we prove that upon the system's non-extinction the ratio between the maximal potential energy and $\log n^2$ converges almost surely to $\frac12$ when $n$ goes to infinity



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