The maximal potential energy of biased random walks on trees

Abstract

The biased random walk on supercritical Galton--Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n{)}^3$, whereas the typical displacement of the walk at the $n$-th step is of order $(\log n{)}^2$. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order $(\log n{)}^2$ in contrast with its typical size, which is of order $\log n$. The proof relies on analyzing the intricate multiscale structure of the potential energy.

arXiv commentpublished in Acta Mathematicae Applicatae Sinica, English Series (2025)

References in corpus (7)

• Convergence in law of the minimum of a branching random walk
• Almost sure convergence for stochastically biased random walks on trees
• Energy dynamics in the Sinai model
• Large deviations for random walks in random environment on a Galton-Watson tree
• Biased random walks on a Galton-Watson tree with leaves
• A proof of the Lyons-Pemantle-Peres monotonicity conjecture for high biases
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Cited by in corpus (3)

• The slow regime of randomly biased walks on trees
• The most visited sites of biased random walks on trees
• Generalized range of slow random walks on trees