Critical large deviations in harmonic crystals with long-range interactions
Let \(P\) be the distribution of a mean zero Gaussian field on \(\mathbb{Z}^d\), \(d\geq 1\), whose covariance is given by the Green function \(G(x,y)= \sum^\infty_{n=0} p^{(n)}(x,y)\) of a random walk \(p(x, y)\). It is assumed that \(p\) is homogeneous, isotropic, symmetric, transient and belongs to the domain of attraction of the symmetric \(\alpha\)-stable law, \(\alpha\in (0,2\wedge d)\). Let \(R_N\) be the empirical field of \(P\) in a cubic box \(\Lambda_N\) of side \(N\in \mathbb{Z}_+\). It is proved that the strong large deviation principle holds for \(R_N\) at the volume scale \(N^{d-\alpha}\) with the rate function given by the Dirichlet form of the embedded \(\alpha\)-stable process. It is also proved that if zero boundary conditions for the Gaussian field outside \(\Lambda_N\) are imposed, then the strong large deviation principle still holds at the same scale but with different rate function. The paper is a continuation of an earlier study of the authors [Commun. Math. Phys. 209, 595-632 (2000; Zbl 0954.60091)] and extends techniques and results of \textit{E. Bolthausen} and \textit{J.-D. Deuschel} [Ann. Probab. 21, 1876-1920 (1993; Zbl 0801.60018)].
scientific article; zbMATH DE number 1905978
scientific article