Determinantal point processes with J-Hermitian correlation kernels

• Autores: Eugene Lytvynov
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 4, 2013, págs. 2513-2543
• Idioma: inglés
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• Resumen

  • Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let ΓX denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on ΓX. A point process μ is called determinantal if its correlation functions have the form k(n)(x1,…,xn)=det[K(xi,xj)]i,j=1,…,n. The function K(x,y) is called the correlation kernel of the determinantal point process μ. Assume that the space X is split into two parts: X=X1⊔X2. A kernel K(x,y) is called J-Hermitian if it is Hermitian on X1×X1 and X2×X2, and K(x,y)=−K(y,x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ for x∈X1 and y∈X2. We derive a necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel K(x,y).