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February 2001 Critical large deviations in harmonic crystals with long-range interactions
P. Caputo, J.-D. Deuschel
Ann. Probab. 29(1): 242-287 (February 2001). DOI: 10.1214/aop/1008956329


We continue our study of large deviations of the empirical measures of a massless Gaussian field on $\mathbb Z^d$, 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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P. Caputo. J.-D. Deuschel. "Critical large deviations in harmonic crystals with long-range interactions." Ann. Probab. 29 (1) 242 - 287, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1021.60022
MathSciNet: MR1825149
Digital Object Identifier: 10.1214/aop/1008956329

Primary: 31C25 , 60F10 , 60G15 , 60G52 , 82B41

Keywords: Dirichlet forms , Gaussian random fields , Gibbs measures , large deviations , Random walks , Symmetric stable processes

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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