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Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 111–148 | Cite as

Determinantal Processes with Number Variance Saturation

  • Kurt JohanssonEmail author
Article

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.

Keywords

Neural Network Statistical Physic Complex System Brownian Motion Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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