RANDOM-WALKS ON ULTRAMETRIC SPACES - MEAN AND VARIANCE OF THE RANGE

We study thermally activated hopping processes on regularly multifurcating ultrametric spaces (UMS) and investigate the first two moments of the range R(n), where n is the number of steps. Paralleling the recent findings for Cayley trees, the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n. For the first moment, the mean number of distinct sites visited, S(n), we derive analytically the cross-over from a linear behaviour in the transient regime gamma > 1 to a sublinear increase S(n) approximately n-gamma in the recurrent regime gamma < 1. Here as usual for UMS, the parameter gamma depends linearly on the temperature. The second moment of R(n), the variance V(n), displays different behaviours corresponding to strongly transient (gamma > 2), not strongly transient (1 < gamma < 2) and recurrent (gamma < 1) random walks. All results are verified by comparison to computer data obtained through simulations and through numerical series expansions.