Exotic local limit theorems in free products of abelian groups
A talk in the Oberseminar Wahrscheinlichkeitstheorie series by
Marc Peigné
| Abstract: | We construct random walks $(X_n)_{n \geq 0}$ on free products of the form $\mathbb{Z}^3 * \mathbb{Z}^d$, with $d = 5$ or $6$ for which the local limit theorem has the following form: $\mu^{*n}(e) \sim CR^{-n} n^{-5/3}$ if $d=5$ and $\mu^{*n}(e) \sim CR^{-n} n^{-3/2} \log (n)^{-1/2}$ if $d=6$ where $\bullet \mu$ is the distribution of the increments of $(X_n)_{n \geq 0}$ $\bullet \mu^{*n}$ is the $n$th convolution power of $\mu$ $\bullet R$ is the inverse of the spectral radius of $\mu$. This disproves a previous result of Candellero and Gilch and shows that the classification of local limit theorems on free products of the form $\mathbb{Z}^{d_1} * \mathbb{Z}^{d_2}$ is incomplete. We will present the context, the tools and the known results and will explain how to choose the distribution $\mu$ to get such "exotic" asymptotics. Joint work with Matthieu Dussaule and Samuel Tapie Within the CRC this talk is associated to the project(s): B10 |