Exponential convergence to equilibrium for a class of random-walk models (Q1118259)

We prove exponential convergence to equilibrium \((L^ 2\) geometric ergodicity) for a random walk with inward drift on a sub-Cayley rooted tree. This random walk model generalizes a Monte Carlo algorithm for the self-avoiding walk proposed by \textit{A. Berretti} and \textit{A. D. Sokal} [ibid. 40, No.3/4, 483-531 (1985)]. If the number of vertices of level N in the tree grows as \(c_ N\sim \mu^ NN^{\gamma -1}\), we prove that the autocorrelation time \(\tau\) satisfies \(<N>^ 2\lesssim \tau \lesssim <N>^{1+\gamma}\).