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.