## Annals of Probability

- Ann. Probab.
- Volume 29, Number 1 (2001), 242-287.

### Critical large deviations in harmonic crystals with long-range interactions

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 29, Number 1 (2001), 242-287.

**Dates**

First available in Project Euclid: 21 December 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1008956329

**Digital Object Identifier**

doi:10.1214/aop/1008956329

**Mathematical Reviews number (MathSciNet)**

MR1825149

**Zentralblatt MATH identifier**

1021.60022

**Subjects**

Primary: 60G15: Gaussian processes 60G52: Stable processes 60F10: Large deviations 31C25: Dirichlet spaces 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

**Keywords**

Random walks Gaussian random fields Gibbs measures large deviations symmetric stable processes Dirichlet forms

#### Citation

Caputo, P.; Deuschel, J.-D. Critical large deviations in harmonic crystals with long-range interactions. Ann. Probab. 29 (2001), no. 1, 242--287. doi:10.1214/aop/1008956329. https://projecteuclid.org/euclid.aop/1008956329