On the asymptotic behaviour of convolution powers of probabilities on discrete groups (Q1114110)

It is shown that there is a discrete countable group \(\Gamma\) and two finitely-supported symmetric probabilities \(\mu_ 1\) and \(\mu_ 2\) on \(\Gamma\) so that both \(Supp(\mu_ 1)\) and \(Supp(\mu_ 2)\) generate \(\Gamma\) and so that \(\mu_ 1^{*n}(e)\sim C_ 1\sigma^ n_ 1/n^{\alpha_ 1}\) and \(\mu_ 2^{*n}(e)\sim C_ 2\sigma^ n_ 2/n^{\alpha_ 2}\) as \(n\to \infty\), where \(0<\sigma_ 1,\sigma_ 2<1\), \(C_ 1,C_ 2>0\) and \(\alpha_ 1,\alpha_ 2>0\), with \(\alpha_ 1\neq \alpha_ 2\). This settles a conjecture of P. Gerl.