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Distributional limits for the symmetric exclusion process. (English) Zbl 1172.60031
In the recent seminal paper by J. Borcea, P. Bränden and T. M. Liggett [”Negative dependence and the geometry of polynomials.” J. Am. Math. Soc. 22, 521–567 (2009)] it is shown that a so called strong Rayleigh property (enjoyed by product measures) is preserved by (evolution of) the symmetric exclusion process $$\eta$$ on a countable set. For the background on $$\eta$$ see Ch.VIII of the monograph [T. M. Ligget, Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften, 276. (New York) etc.: Springer-Verlag. (1985; Zbl 0559.60078)]. Using this fact the author proves convergence to Poisson and Gaussian laws for functionals (in partial sums) of the process $$\eta$$ by establishing bounds for covariances. One benefits from the coincidence of distributions of $$\sum_{i\leq n}\eta_i, n\in\mathbb N,$$ under a strong Rayleigh probability measure $$\mu$$ on $$\{0,1\}^n$$, with those of sums of $$n$$ independent Bernoulli variables.
Note that the strong Rayleigh property, equivalent to stability of generating polynomial for $$\mu$$, entails negative association and other related properties. An auxiliary result implying preservation of stability by $$\eta$$ is proved as well.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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##### References:
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