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

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].

For the entire collection see [Zbl 0852.00016].

Reviewer: K.Schürger (Bonn)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |