The maximal potential energy of biased random walks on trees

2014/03/26 by Yueyun Hu, Hu, Yueyun, Zhan Shi +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1403.6799

openalex publication_date 2014/03/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• A proof of the Lyons-Pemantle-Peres monotonicity conjecture for high biases

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