Abstract
This Letter gives detailed proofs concerning the analysis of the pair correlations for a nonconvex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operator. Although the problem is similar to the semiclassical study of the Kac operator presented in a paper with M. Brunaud, which was devoted⋆⋆ to the study of exp-(v/2) exp h 2 Δ exp-(v/2) for h small, new features appear for the model exp-(v/2h) exp hΔ exp-(v/2h). Our principal results concern the splitting of this operator between the two largest eigenvalues. We give an upper and a lower bound for this splitting in the semi-classical regime. As a corollary, we get good control of the decay of the pair correlation. Some of the results were announced in a previous paper. Related WKB constructions will be developed in a later paper.
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Inspired by papers by M. Kac [15, 16].
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Helffer, B. On laplace integrals and transfer operators in large dimension: Examples in the nonconvex case. Lett Math Phys 38, 297–312 (1996). https://doi.org/10.1007/BF00398354
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DOI: https://doi.org/10.1007/BF00398354