The Simple Exclusion Process

Abstract

Among the simplest and most widely studied interacting particle systems is the simple exclusion process. It represents the evolution of random walks on the lattice ℤ^d with a hard-core interaction which prevents more than one particle per site. We apply in this chapter the results presented in the first part of the book to additive functionals of exclusion processes. In Sect. 5.2 we prove this result in the case where the particles evolve according to symmetric random walks, in which case the Bernoulli measures are reversible. In Sect. 5.3, we show that the generator of the exclusion process associated to random walks whose transition probability has mean zero satisfies a sector condition. A central limit theorem follows by the reasons presented in Sect. 2.7.3. In Sects. 5.4, 5.5 and 5.6, we examine the case where the transition probability has a non-vanishing mean. In this case the L ² spaces associated to the stationary states of the exclusion process can be decomposed as a direct sum of orthogonal spaces. In dimension d≥3, the generator of the process satisfies the graded sector conditions stated in Sect. 2.7.4. This estimate provides a central limit theorem for local functions V in ${{\mathcal{H}}_{-1}}$. In the last section of this chapter we state some results on transient Markov processes, needed to prove the graded sector condition.