TY - JOUR

T1 - Determinantal Point Processes and Fermion Quasifree States

AU - Olshanski, Grigori

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85081738443&partnerID=8YFLogxK

U2 - 10.1007/s00220-020-03716-1

DO - 10.1007/s00220-020-03716-1

M3 - Article

AN - SCOPUS:85081738443

VL - 378

SP - 507

EP - 555

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -