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Ornstein-Zernike asymptotics for a general “two-particle” lattice operator. (English) Zbl 1235.82025
The paper is devoted to the study of the asymptotic behavior of correlations for a general “two-particle” operator \(T\) acting on the Hilbert space \(\ell_2(\mathbb{Z}^d\times \mathbb{Z}^d)\), for all dimension \(d = 1, 2,\dots.\) The operator \(T\) is presented as the sum of a “main” term, and a small “interacting” term, a form which appears in many problems. It is proved that, if the interacting term is small, then a complete description of the asymptotics for large \(t\) of the correlations \((T^t f^{(1)},f^{(2)})\), \(t = 1, 2, \dots\), for \(f^{(1)}, f^{(2)}\) in some suitable class. The obtained asymptotics is exponential with a power-law factor, which is \(t^{-d}\) for \(d\geq 3\), but, for \(d = 1, 2\), it can be “anomalous” and is determined by the interacting term. Such a behaviour is denoted by Ornstein-Zernike type. One of the main tools to prove the results is a deep analysis of the spectrum of \(T\).

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60K37 Processes in random environments
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