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<p>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 <i>n</i> steps is of order (log <i>n</i>)<sup>3</sup>, whereas the typical displacement of the walk at the <i>n</i>-th step is of order (log <i>n</i>)<sup>2</sup>. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order (log <i>n</i>)<sup>2</sup> in contrast with its typical size, which is of order log <i>n</i>. The proof relies on analyzing the intricate multiscale structure of the potential energy.</p>

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The Maximal Potential Energy of Biased Random Walks on Trees

  • 期刊论文

  • 发表日期：

    2025年6月24日

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  • Yueyun Hu,
  • Zhan Shi

摘要

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.