Abstract
We show that the $\operatorname{Sine}_{\beta}$ point process, defined as the scaling limit of the Circular Beta Ensemble when the dimension goes to infinity, and generalizing the determinantal sine-kernel process, is rigid in the sense of Ghosh and Peres: the number of points in a given bounded Borel set $B$ is almost surely equal to a measurable function of the position of the points outside $B$.
Highlights
If E is a complete separable metric space, we can define a point process on E as a random purely atomic Radon measure X on E, which can be viewed as a random locally finite collection of points in M, with possible repetitions, the order of the points being irrelevant
It has been observed that some point processes satisfy the unusual property that the number of points in any Borel set is uniquely determined by the points in the complement of this set
A point process X on a complete separable metric space E is rigid if and only if for all bounded Borel subsets B of E, the number of points X(B) in B is measurable with respect to the σ-algebra ΣE\B
Summary
If E is a complete separable metric space, we can define a point process on E as a random purely atomic Radon measure X on E, which can be viewed as a random locally finite collection of points in M , with possible repetitions, the order of the points being irrelevant. The determinantal sine-kernel is a real-valued point process (E = R), defined as follows: it has no multiple points and its m-point correlation function at x1, . Some two-dimensional point processes are proven to be rigid, including the infinite Ginibre ensemble and the set of zeros of some Gaussian analytic functions (see Ghosh and Peres [GP17]).
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