The Maximal Potential Energy of Biased Random Walks on Trees

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.