The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on Z

• Published in: The Annals of probability Vol. 11; no. 2; pp. 362 - 373
• Main Author: Arratia, Richard
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics 01-05-1983; The Institute of Mathematical Statistics
• Subjects: 60K35; Brownian motion; correlation inequalities; Correlations; Integers; Interacting particle system; Markov processes; Particle collisions; Particle interactions; Particle motion; Quantum statistics; random permutations; Random walk; simple exclusion process; Statistical theories
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1176993602

Consider a system of particles moving on the integers with a simple exclusion interaction: each particle independently attempts to execute a simple symmetric random walk, but any jump which would carry a particle to an already occupied site is suppressed. For the system running in equilibrium, we analyze the motion of a tagged particle. This solves a problem posed in Spitzer's 1970 paper "Interaction of Markov Processes." The analogous question for systems which are not one-dimensional, nearest-neighbor, and either symmetric or one-sided remains open. A key tool is Harris's theorem on positive correlations in attractive Markov processes. Results are also obtained for the rightmost particle in the exclusion system with initial configuration Z-, and for comparison systems based on the order statistics of independent motions on the line.