## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 2 (1983), 362-373.

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

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 2 (1983), 362-373.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993602

**Digital Object Identifier**

doi:10.1214/aop/1176993602

**Mathematical Reviews number (MathSciNet)**

MR690134

**Zentralblatt MATH identifier**

0515.60097

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Interacting particle system simple exclusion process random permutations correlation inequalities

#### Citation

Arratia, Richard. The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on $Z$. Ann. Probab. 11 (1983), no. 2, 362--373. doi:10.1214/aop/1176993602. https://projecteuclid.org/euclid.aop/1176993602