Determinantal Processes with Number Variance Saturation

By K. Johansson. 2004

43 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Non-intersecting random walks [Johansson, 2004] Consider a set of k independent, simple, symmetric random walks of length T on the integers.”
Hyperuniform states of matter (2018, background): “Examples of determinantal point processes arise in uniform spanning trees [228], self-avoiding random walks [229] and random polynomials [230], and have been reported both on the real line [34, 231–233] and in higher Euclidean space dimensions [34, 136, 233].”
Negative dependence and the geometry of polynomials (2007, background): “[5, 57], such measures have important applications ranging from fermionic processes/continuous scaling limits of discrete point processes [23, 59] to the distribution of eigenvalues of random matrices and zeros of the Riemann zeta function [21], transfer current matrices [16], non-intersecting rando”
Directed polymers and the quantum Toda lattice (2009, background): “In particular, it is a determinantal point process [29].”
Determinantal Processes and Independence (2005, background): “Determinantal processes arise naturally in several other settings, including eigenvalues of random matrices, random spanning trees and nonintersecting paths; see, e.g., Burton and Pemantle [4], Soshnikov [27], Lyons [20], Lyons and Steif [21], Shirai and Takahashi [25], Johansson [14], Borodin, Okou”