Ornstein-Zernike Asymptotics for a General “Two-Particle” Lattice Operator

Abstract

We study the asymptotic behavior of correlations for a general “two-particle” operator ${{\mathcal T}}$ acting on the Hilbert space ${\ell_2({\mathbb Z}^d\times {\mathbb Z}^d)}$, for all dimension d = 1, 2, . . .. ${{\mathcal T}}$ is written as the sum of a “main” term, and a small “interacting” term, a form which appears in many problems. If the interacting term is small, we give a complete description of the asymptotics for large t of the correlations ${({\mathcal T}^{t} f^{(1)}, f^{(2)}), t=1,2,\ldots}$, for f ⁽¹⁾, f ⁽²⁾ in some suitable class. The asymptotics is of the Ornstein-Zernike type, i.e., exponential with a power-law factor, which is t ^−d for d ≥ 3, but for d = 1, 2 it can be “anomalous” and is determined by the interacting term.