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.
Publié le : 2001-02-14
Classification:
Random walks,
Gaussian random fields,
Gibbs measures,
large deviations,
symmetric stable processes,
Dirichlet forms,
60G15,
60G52,
60F10,
31C25,
82B41
@article{1008956329,
author = {Caputo, P. and Deuschel, J.-D.},
title = {Critical large deviations in harmonic crystals with long-range
interactions},
journal = {Ann. Probab.},
volume = {29},
number = {1},
year = {2001},
pages = { 242-287},
language = {en},
url = {http://dml.mathdoc.fr/item/1008956329}
}
Caputo, P.; Deuschel, J.-D. Critical large deviations in harmonic crystals with long-range
interactions. Ann. Probab., Tome 29 (2001) no. 1, pp. 242-287. http://gdmltest.u-ga.fr/item/1008956329/