Abstract
We prove tail triviality of determinantal point processes mu on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for hbox {Sine}_2 , hbox {Airy}_2 , hbox {Bessel}_2 , and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.
Highlights
Let S be a locally compact, complete, separable metric space with metric d(·, ·)
To prove tail triviality we introduce a discrete approximation for determinantal point processes, called the tree representation
The associated determinantal point process μGin is called the Ginibre point process
Summary
Let S be a locally compact, complete, separable metric space with metric d(·, ·). We assume S is unbounded. To prove tail triviality we introduce a discrete approximation for determinantal point processes, called the tree representation. Because μK is a determinantal point process on the discrete space, its tail σ -field is trivial. Such weak convergence, does not suffice for the convergence of the values on the tail σ -field Tail(S). Bufetov has proved independently tail triviality of determinantal point processes on continuous spaces independently of us (a seminar talk at Kyushu University in October 2015). His method is completely different from ours and requires a restriction on an integrability condition of the determinantal kernel K(x, y).
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