Algebraic $L^2$ Decay of Attractive Critical Processes on the Lattice

Abstract

We consider a special class of attractive critical processes based on the transition function of a transient random walk on $\mathbb{Z}^d$. These processes have infinitely many invariant distributions and no spectral gap. The exponential $L^2$ decay is replaced by an algebraic $L^2$ decay. The paper shows the dependence of this algebraic rate in terms of the dimension of the lattice and the locality of the functions under consideration. The theory is illustrated by several examples dealing with locally interacting diffusion processes and independent random walks.