A local limit theorem for convergent random walks on relatively hyperbolic groups

Abstract

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius. When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form pn(e, e) ∼ CR −n n −d/2 , where pn(e, e) is the probability of returning to the origin at time n, R is the inverse of the spectral radius of the random walk and d is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate. This concludes the classification all possible behaviour for pn(e, e) on such groups.

Local limit theorems in relatively hyperbolic groups I: rough estimates

“…where µ * n is the nth convolution power of µ and R is the inverse of the spectral radius of µ. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a first version of [11]. This also shows that the classification of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.…”

“…Our main goal in this note is to disprove [7,Lemma 4.5] and a similar statement that appeared in a first version of [11]. In particular, we prove that the classification obtained in [7] is incomplete: we derive a local limit theorem on Z 3 * Z 5 of the form (1.1) but with unexpected exponent α = 5/3, and a local limit theorem on Z 3 * Z 6 which is not of the form (1.1).…”

“…A more general statement also appeared in a first version of [11], which led the authors to modify their statement. Define for i = 1, 2,…”

“…In particular, they proved that they always are of the form (1.1), with α = 3/2 or α = d i /2 and the latter case can only happen if d i ≥ 5. Although in this paper we will not work in the general setting of relatively hyperbolic groups, let us mention that free products are the simplest examples of such groups and results of [7] are being generalized to this setting in recent works by the authors, see [8], [9], [11].…”

“…When the random walk is spectrally positive recurrent, we can only prove the rough estimate µ * n (e) ≍ R −n n −3/2 , which means that the ratio of the quantities on the left and right hand-side is bounded away from 0 and infinity [8]. When the random walk is convergent and parabolic subgroups are virtually abelian, the local limit theorem has the form (1.1), with α = d/2, where d is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate [11]. Moreover, in this situation, one can only have d ≥ 5.…”