Speaker: Yahui Qu (University of Wisconsin Madison)
Title: Pair correlation function of the Sine-beta Process
Language: English
Time & Venue: 2026 年6月8日16:00–17:00 南楼613
Abstract: In Random Matrix Theory, a central problem is to understand the limiting behavior of eigenvalues as the matrix size tends to infinity. For a broad class of matrix models-beta ensembles, the bulk scaling limit is the Sine beta point process. A fundamental description of a point process is through its correlation function. In the classical cases (beta equals 1, 2, and 4), the correlation functions can be expressed using explicit formulas due to the special structure of the point process. For general beta, however, no such representation is available, and describing even the two-point correlation function has remained a long-standing open problem. In this talk, I will present a representation for the two-point correlation function of the Sine beta process. The representation works for all beta>0, and expresses the pair correlation in terms of moments of a certain random variable, and also in terms of an infinite system of ODEs. This characterization allows us to recover the known formulas for beta equal to 2 and 4. In particular, we can present a new formula for beta equal to 6. Moreover, this representation allows us to establish continuity properties of the pair correlation function and to obtain estimates on its asymptotic decay. Joint work with Benedek Valkó.
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