DETERMINANTAL POINT PROCESSES WITH J-HERMITIAN CORRELATION KERNELS

作者：Lytvynov Eugene^*

来源：Annals of Probability, 2013, 41(4): 2513-2543.

DOI：10.1214/12-AOP795

Let X be a locally compact Polish space and let in 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), ... , x(n)) = det[K (x(i), x(j))](i, j=1, ... ,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 boolean OR X-2. A kernel K (x, y) is called J-Hermitian if it is Hermitian on X-1 x X-1 and X-2 x X-2, and K (x, y) = -%26lt;(K(y, x))over bar%26gt; for x is an element of X-1 and y is an element of 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).

• 出版日期2013-7