×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arratia, R., The motion of a tagged particle in the simple symmetric exclusion process on \(Z\), Ann. probab., 11, 362-373, (1983) · Zbl 0515.60097
[2] Barbour, A.D.; Holst, L.; Janson, S., Poisson approximation, (1992), Oxford University Press · Zbl 0746.60002
[3] J. Borcea, P. Brändén, Linear operators preserving stability, 2008
[4] J. Borcea, P. Brändén, T.M. Liggett, Negative dependence and the geometry of polynomials, 2008
[5] Brändén, P., Polynomials with the half-plane property and matroid theory, Adv. math., 216, 302-320, (2007) · Zbl 1128.05014
[6] Bulinski, A.; Shashkin, A., Limit theorems for associated random fields and related systems, (2007), World Scientific · Zbl 1154.60037
[7] Cartwright, D.I.; Woess, W., Infinite graphs with nonconstant Dirichlet finite harmonic functions, SIAM J. disc. math., 5, 380-385, (1992) · Zbl 0752.31005
[8] De Masi, A.; Ferrari, P.A., Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process, J. stat. phys., 107, 677-683, (2002) · Zbl 1008.82022
[9] Jara, M.D.; Landim, C., Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion, Ann. I. H. Poincaré, 42, 567-577, (2006) · Zbl 1101.60080
[10] Kipnis, C.; Varadhan, S.R.S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. math. phys., 104, 1-19, (1986) · Zbl 0588.60058
[11] Liggett, T.M., Interacting particle systems, (1985), Springer-Verlag · Zbl 0559.60078
[12] C.M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability, IMS, 1984, pp. 127-140
[13] M. Peligrad, S. Sethuraman, On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion, 2008 · Zbl 1162.60347
[14] Pemantle, R., Towards a theory of negative dependence, J. math. phys., 41, 1371-1390, (2000) · Zbl 1052.62518
[15] Roussas, G.G., Asymptotic normality of random fields of positively or negatively associated processes, J. multivariate anal., 50, 152-173, (1994) · Zbl 0806.60040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.