Exact clesed form of the return probability on the Bethe lattice

TL;DR

This paper derives an exact closed-form expression for the return probability of a random walk on the Bethe lattice, confirming previous asymptotic results and clarifying the behavior in degenerate cases.

Contribution

It provides the first exact closed-form solution for the return probability on the Bethe lattice, resolving discrepancies with asymptotic approximations.

Findings

Exact return probability formula derived

Asymptotic form confirmed and clarified

Density of states calculated

Abstract

An exact closed form solution for the return probability of a random walk on the Bethe lattice is given. The long-time asymptotic form confirms a previously known expression. It is however shown that this exact result reduces to the proper expression when the Bethe lattice degenerates on a line, unlike the asymptotic result which is singular. This is shown to be an artefact of the asymptotic expansion. The density of states is also calculated.