, 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.

Communicated by P. Sarnak
Dedicated to Freeman J. Dyson on his 80 th birthday
Supported by the Swedish Science Research Council and the Göran Gustafsson Foundation (KVA).