Random walks on Cayley trees as models for relaxation in a hierarchical system

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Abstract
We consider a nearest-neighbor random walk on a finite Cayley tree with hopping rates depending on a level index in an arbitrary way, and construct an exact formula for the Laplace transform of the first passage time from one point to another. This function determines, by the renewal equation, the propagator and hence all the properties of the solution. Finally, we apply the formalism to a simple case of thermally activated hopping, and show that the time of return to the origin has, to a good degree of approximation, a stable distribution.

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