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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.

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Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Kurt Johansson

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  1. Kurt Johansson
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Correspondence to Kurt Johansson.

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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).

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Johansson, K. Determinantal Processes with Number Variance Saturation. Commun. Math. Phys. 252, 111–148 (2004). https://doi.org/10.1007/s00220-004-1186-4

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