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.