Pietro Caputo and Jean-Dominique Deuschel
Critical large deviations in 
harmonic crystals with long range interactions
(388K, gzipped postscript file)

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.