Interacting random walk of two particles in a dynamical random environment. Decay of correlations.

This paper continues the study of a family of models studied earlier by the authors. Two particles perform discrete-time symmetric random walks on the d-dimensional integer lattice Z^d and interact locally with each other and with a random field (the “environment”) which is indexed by the lattice points. The environment evolves randomly in time its law of evolution is locally affected by the particles . The whole system is Markovian, and all interactions are assumed to be sufficiently small. It is shown that the correlations of the field at two fixed points decay in time as C t^{(−d/2)−1}. Under additional assumptions the constant C may be expressed to first order as the sum of the corresponding constants for the one-particle model. The proofs are based on the analysis of the spectrum of the system’s transition operator. The results may be extended to models containing a finite number of particles