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arXiv · 2303.05818

Exotic local limit theorems at the phase transition in free products

Matthieu Dussaule·Marc Peigné·Samuel Tapie
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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 $\mu$ * n (e) $\sim$ CR --n n --5/3 if d = 5 and $\mu$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if d = 6, where $\mu$ * n is the nth convolution power of $\mu$ and R is the inverse of the spectral radius of $\mu$. 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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Matthieu Dussaule, Marc Peigné, Samuel Tapie. 2023-03-10. Exotic local limit theorems at the phase transition in free products. https://arxiv.org/abs/2303.05818

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