Volume 8 (2025)
Dussaule, Matthieu;  Wang, Longmin;  Yang, Wenyuan
The growth of the Green function for random walks and Poincaré series
Annales Henri Lebesgue, Volume 8 (2025), pp. 1061-1107

Metadata

DOI10.5802/ahl.256
Keywords Branching random walks ,  relatively hyperbolic group ,  growth rate ,  Green function ,  Poincaré series ,  Growth tightness ,  Patterson–Sullivan measure ,  parabolic gap

Abstract

Given a probability measure $\mu $ on a finitely generated group $\Gamma $, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu $. Endowing $\Gamma $ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu $ on $\Gamma $. Our main result exhibits a relatively hyperbolic group with convergent Poincaré series generated by $H_r(n)$.


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