00-102 Pietro Caputo and Jean-Dominique Deuschel

Critical large deviations in harmonic crystals with long range interactions (388K, gzipped postscript file) Mar 7, 00

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Abstract. We continue our study of large deviations of the empirical measures of a massless Gaussian field on $\bbZ^d$, whose covariance is given by the Green function of a long range random walk, \cite{CD}. In this paper we extend techniques and results of \cite{BD} to the {\em non-local} 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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