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Preprints, Working Papers, ... Year : 2023

Exotic local limit theorems at the phase transition in free products

Abstract

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that µ * n (e) ∼ CR −n n −5/3 if d = 5 and µ * n (e) ∼ CR −n n −3/2 log(n) −1/2 if d = 6, 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 rst version of [11]. This also shows that the classication 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.
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Dates and versions

hal-04017794 , version 1 (07-03-2023)
hal-04017794 , version 2 (09-03-2023)

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Matthieu Dussaule, Marc Peigné, Samuel Tapie. Exotic local limit theorems at the phase transition in free products. 2023. ⟨hal-04017794v2⟩
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