Extremal eigenvalue fluctuations in the GUE minor process and the law of fractional logarithm

2015/05/21 by Paquette, Elliot, Zeitouni, Ofer
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1505.05627

Abstract

We consider the GUE minor process, where a sequence of GUE matrices is drawn from the corner of a doubly infinite array of i.i.d. standard normal variables subject to the symmetry constraint. From each matrix, we take its largest eigenvalue, appropriately rescaled to converge to the standard Tracy-Widom distribution. We show the analogue of the law of iterated logarithm for this sequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factor and show the limsup of this sequence is a constant almost surely. We also give almost sure bounds for the appropriately scaled liminf.

Related

• The Paquette-Zeitouni law of fractional logarithms for the GUE minor process and the Plancherel growth process
• Law of fractional logarithm for random matrices
• On the distribution of eigenvalues of GUE and its minors at fixed index
• Global spectral fluctuations in the Gaussian Unitary Ensemble
• Fluctuations at the edges of the spectrum of the full rank deformed GUE