Universality for conditional measures of the sine point process

Abstract

The sine process is a rigid point process on the real line, which means that for almost all configurations X, the number of points in an interval I = [−R, R] is determined by the points of X outside of I. In addition, the points in I are an orthogonal polynomial ensemble on I with a weight function that is determined by the points in X \I. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length |I| = 2R tends to infinity, thereby answering a question posed by A.I. Bufetov.

DLR Equations and Rigidity for the Sine‐Beta Process

“…The answer to this question is positive when h 2, as a consequence of the work [32] where, for Sine 2 a.e. P Conf.R/, the asymptotic of the conditional measure c .x 1 ; : : : ; x N / has been shown to be Sine 2 in the limit where h R; R and R 3 I.…”

Universality for conditional measures of the Bessel point process

“…First we will show, in Section 4.6, that the asymptotics of the approximating weights actually imply (4.9) and hence Theorem 1.1. For this, we use techniques introduced by Lubinsky in [16] that are also used in [13].…”

“…where in the last step we used the computation of the equilibrium measure of the external field that plays a role in [13].…”

Large gap asymptotics for the generating function of the sine point process

“…The conditional measure of the sine process on admits an explicit density, see [13, Theorems 1.1 and 1.4]. Furthermore, the correlation kernel of this conditional sine process converges to the usual sine kernel as the size of the interval gets large, see [45, Theorems 1.3 and 1.4].…”