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Regular version of the site

Article

Determinantal Point Processes and Fermion Quasifree States

Communications in Mathematical Physics. 2020. Vol. 378. No. 1. P. 507-555.

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 (CommunMath Phys 353:853–903,
2017). As an application we resolve the equivalence/disjointness dichotomy for some
of those processes.