Exotic local limit theorems at the phase transition in free products (Q6595699)

Let \(\Gamma\) be a finitely generated group and let \(\mu\) be a finitely supported and symmetric probability measure on \(\Gamma\). Let \(\mu^{\ast n}\) be the \(n\)th convolution power of \(\mu\):\N\[\N\mu^{\ast n}(x)=\sum_{y_{1}, \ldots,y_{n-1} \in \Gamma} \mu(y_{1})\mu(y_{1}^{-1}y_{2})\ldots \mu(y_{n-1}^{-1}x).\N\]\NLet \((X_{n})_{n}\) be a \(\mu\)-random walk starting at the identity element \(e \in \Gamma\) is defined by \(X_{n}= g_{1}\ldots g_{n}\) where \(g_{k}\) are independent random variables whose distribution is given by \(\mu\). Then \(\mu^{\ast n}\) is the \(n\)th step distribution of the random walk, so for all \(x \in \Gamma\), \(\mu^{\ast n}(x)\) is the probability that \(X_{n}=x\). The spectral radius of the random walk is defined by \(\rho=\lim \sup \mu^{\ast n}(x)(x)^{1/2} \in [0,1]\) and is independent of \(x\), provided that \(\mu\) is admissible (see [\textit{W. Woess}, Random walks on infinite graphs and groups. Cambridge: Cambridge University Press (2000; Zbl 0951.60002)]).\N\NThe local limit problem consists in finding the asymptotic behavior of \(\mu^{\ast n}(x)\) as \(n\) goes to infinity (assuming, for simplicity, that \(\mu\) is aperiodic, i.e., there exists \(n_{0}\) such that for every \(n \geq n_{0}\), \(\mu^{\ast n}(e)>0)\).\N\NIn the paper under review, the authors construct random walks on free products of the form \(\mathbb{Z}^{3} \ast \mathbb{Z}^{d}\), with \(d \in \{5,6\}\) which are divergent and not spectrally positive recurrent. They then derive a local limit theorem for these random walks, proving that \(\mu^{\ast n}(e) \sim C\rho^{n} n^{-3/2}\) if \(d=5\) and \(\mu^{\ast n}(e) \sim C\rho^{n}n^{-3/2} \log(n)^{-1/2}\) if \(d = 6\). This disproves a result of \textit{E. Candellero} and \textit{L. Gilch} [Random Struct. Algorithms 40, No. 2, 150--181 (2012; Zbl 1242.05251), Lemma 4.5]. As a consequence, 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.