Exotic local limit theorems at the phase transition in free products

Abstract

We construct random walks on free products of the form ${\mathbb{Z}}^3\ast {\mathbb{Z}}^d$, 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 ${\mathrm{\mu }}^{\ast n}(e)\sim C{R}^{-n}{n}^{-5∕3}$ if $d=5$ and ${\mathrm{\mu }}^{\ast n}(e)\sim C{R}^{-n}{n}^{-3∕2}log{(n)}^{-1∕2}$ if $d=6$, where ${\mathrm{\mu }}^{\ast n}$ 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 [12]. This also shows that the classification of local limit theorems on free products of the form ${\mathbb{Z}}^{d_1}\ast {\mathbb{Z}}^{d_2}$ or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.