, Volume 252, Issue 1-3, pp 111-148
Date: 17 Sep 2004

Determinantal Processes with Number Variance Saturation

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Abstract

Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N→∞. This limting determinantal process has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann ζ-function, [21, 2]. The process can also be constructed using non-intersecting paths and we consider several variants of this construction. One construction leads to a model which shows a transition from a non-universal behaviour with number variance saturation to a universal sine-kernel behaviour as we go up the line.