The complex story of simple exclusion. (English) Zbl 0866.60092

Varadhan, S. R. S.

Ikeda, N. (ed.) et al., Itô’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Itô on the occasion of his 80th birthday. Tokyo: Springer. 385-400 (1996).

In this expository article the author is dealing with the simple exclusion model which perhaps is the simplest model of an interacting particle system on \(Z^d\). Here, the jump size \(z\) is randomly chosen from \(Z^d\) according to a probability distribution \(p(z)\), i.e., if the particle under consideration is located at \(x\), and \(z\) is chosen as the jump size, the jump is executed provided site \(x+z\) is unoccupied. If, however, site \(x+z\) is occupied, the jump is disallowed and the particle starts waiting for a new exponential time. The author considers the following three classes of models: (I) \(p\) is symmetric, i.e. \(p(z)= p(-z)\). (II) \(p\) is asymmetric but has finite expectation \(b=0\). (III) \(p\) has finite expectation \(b\neq 0\). In order to simplify the presentation, the author considers a finite periodic lattice \(Z_N^d\) of integers modulo \(N\) (in every coordinate) where \(N\) is a large integer, and \(p\) is assumed to be local, i.e. \(p(z) =0\) for \(z\notin F\) \((F\) being some fixed finite set). For each of the above classes, some asymptotic results are presented which involve hydrodynamic scaling.
For the entire collection see [Zbl 0852.00016].

Reviewer: K.Schürger (Bonn)

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

expository article; simple exclusion model; interacting particle system; hydrodynamic scaling