The Annals of Probability

Critical large deviations in harmonic crystals with long-range interactions

P. Caputo and J.-D. Deuschel

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We continue our study of large deviations of the empirical measures of a massless Gaussian field on $Z^d$, whose covariance is given by the Green function of a long-range random walk. In this paper we extend techniques and results of Bolthausen and Deuschel to the nonlocal case of a random walk in the domain of attraction of the symmetric $\alpha$-stable law, with $\alpha \in (0, 2 \wedge d)$. In particular, we show that critical large deviations occur at the capacity scale $N^{d-\alpha}$, with a rate function given by the Dirichlet form of the embedded $\alpha$-stable process. We also prove that if we impose zero boundary conditions, the rate function is given by the Dirichlet form of the killed $\alpha$- stable process.

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Ann. Probab., Volume 29, Number 1 (2001), 242-287.

First available in Project Euclid: 21 December 2001

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G52: Stable processes 60F10: Large deviations 31C25: Dirichlet spaces 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Random walks Gaussian random fields Gibbs measures large deviations symmetric stable processes Dirichlet forms


Caputo, P.; Deuschel, J.-D. Critical large deviations in harmonic crystals with long-range interactions. Ann. Probab. 29 (2001), no. 1, 242--287. doi:10.1214/aop/1008956329.

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