Open Access
July 2013 Determinantal point processes with $J$-Hermitian correlation kernels
Eugene Lytvynov
Ann. Probab. 41(4): 2513-2543 (July 2013). DOI: 10.1214/12-AOP795


Let $X$ be a locally compact Polish space and let $m$ be a reference Radon measure on $X$. Let $\Gamma_{X}$ 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 $\Gamma_{X}$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n)}(x_{1},\ldots,x_{n})=\det[K(x_{i},x_{j})]_{i,j=1,\ldots,n}$. The function $K(x,y)$ is called the correlation kernel of the determinantal point process $\mu$. Assume that the space $X$ is split into two parts: $X=X_{1}\sqcup X_{2}$. A kernel $K(x,y)$ is called $J$-Hermitian if it is Hermitian on $X_{1}\times X_{1}$ and $X_{2}\times X_{2}$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_{1}$ and $y\in X_{2}$. We derive a necessary and sufficient condition of existence of a determinantal point process with a $J$-Hermitian correlation kernel $K(x,y)$.


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Eugene Lytvynov. "Determinantal point processes with $J$-Hermitian correlation kernels." Ann. Probab. 41 (4) 2513 - 2543, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1306.60053
MathSciNet: MR3112924
Digital Object Identifier: 10.1214/12-AOP795

Primary: 47B50 , 47G10 , 60K35
Secondary: 45B05 , 46C20

Keywords: $J$-self-adjoint operator , determinantal point process , Fredholm determinant

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • July 2013
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