Ergodic theorems for coupled random walks and other systems with locally interacting components

Abstract

In [5] the second author introduced a variety of new infinite systems with locally interacting components. On the basis of computations for the finite analogues of these systems, he made conjectures ragarding their limiting behavior as t→∞. This paper is devoted to the construction of these processes and to the proofs of these conjectures. We restrict ourselves primarily to spatially homogeneous situations; interesting problems remain unsolved in inhomogeneous cases. Two features distinguish these processes from most other infinite particle systems which have been studied. One is that the state spaces of these systems are noncompact; the other that even though the invariant measures are not generally of product form, one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium. The second moment computations are of inherent interest of course, and they play an important role in the proofs of the ergodic theorems as well.