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We shall consider Euclidean stationary point processes which have fast decay of correlations i.e., their correlation functions factorize upto an additive error decaying exponentially in the separation distance. By a quasi-local statistic of the point process, we refer to statistics that can be expressed as sum of contributions from the points and the contribution of every point being determined by a random ball around the point whose radius has an exponential tail. There are many well-known point processes and statistics that satisfy these conditions and I will present a generic central limit theorem for the same assuming some ?verifiable? moment conditions and a ?non-trivial? variance lower bound. This is based on a joint work with B. Blaszczyszyn and J. E. Yukich. We will then discuss a similar generic central limit theorems for point processes on discrete Cayley graphs with polynomial growth. These results apply for quasi-local statistics of many off-critical spin models on such graphs. For example, the off-critical Ising model and level sets of the Massive Gaussian free field. This is joint work with T. R. Reddy and S. Vadlamani. In view of these results and more results on asymptotic normality for statistics of other random graph models, one is lead to ask the same question about local or quasi-local statistics on unimodular random graphs.

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• Probabilités, Statistiques mathématiques, Mathématiques appliquées (519)

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