Phase Transitions for Random Walk Asymptotics on Free Products of Groups

Suppose we are given finitely generated groups ${\mathrm{\Gamma }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\Gamma }}_m$ equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product {${\mathrm{\Gamma }}_1\ast \mathrm{.}\mathrm{.}\mathrm{.}\ast {\mathrm{\Gamma }}_m$} and give a complete classification of the possible asymptotic behaviour of the corresponding $n$-step return probabilities. They either inherit a law of the form ${\varrho }^{n\delta }{n}^{-{\lambda }_i}{\log }^{{\kappa }_i}n$ from one of the free factors ${\mathrm{\Gamma }}_i$ or obey a ${\varrho }^{n\delta }{n}^{-3\mathrm{/}2}$-law, where $\varrho <1$ is the corresponding spectral radius and $\delta$ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form ${\mathbb{Z}}^{d_1}\ast \mathrm{.}\mathrm{.}\mathrm{.}\ast {\mathbb{Z}}^{d_m}$. Moreover, we characterize the possible phase transitions of the non-exponential types $n^{-{\lambda }_i}{\log }^{{\kappa }_i}n$ in the case ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$.