Asymptotics of the transition probabilities of the simple random walk on self-similar graphs

Abstract

It is shown explicitly how self-similar graphs can be obtained as `blow-up' constructions of finite cell graphs $\hat C$. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph $\hat C$, starting in a vertex $v$ of the boundary of $\hat C$. It is proved that the expected number of returns to $v$ before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the $n$-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the $n$-step transition probabilities.