Local limit theorems in relatively hyperbolic groups I : rough estimates

Matthieu Dussaule (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive-recurrence for random walks on relatively hy-perbolic groups. We then use our estimates for the Green function to prove that pn ≍ R −n n −3/2 for spectrally positive-recurrent random walks, where pn is the probability of going back to the origin at time n and where R is the spectral radius of the random walk.

Domaines: Probabilités [math.PR]; Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]

Matthieu Dussaule. Local limit theorems in relatively hyperbolic groups I : rough estimates. 2020. ⟨hal-02554232⟩