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The potential energy of biased random walks on trees

Abstract : 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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https://hal.archives-ouvertes.fr/hal-00962241
Contributor : Yueyun Hu <>
Submitted on : Tuesday, April 12, 2016 - 10:53:38 PM
Last modification on : Tuesday, May 5, 2020 - 1:03:21 PM
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  • HAL Id : hal-00962241, version 3
  • ARXIV : 1403.6799

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Yueyun Hu, Zhan Shi. The potential energy of biased random walks on trees. 2014. ⟨hal-00962241v3⟩

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