Determinantal Point Processes and Fermion Quasifree States

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2 Citations (Scopus)


Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes. We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations. We prove that the formalism applies to discrete N-point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Borodin and Olshanski (Commun Math Phys 353:853–903, 2017). As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

Original languageEnglish
Pages (from-to)507-555
Number of pages49
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - 1 Aug 2020


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