Gaussian fluctuations of eigenvalues in the GUE

Abstract

Under certain conditions on k we calculate the limit distribution of the k:th largest eigenvalue, x_k, of the Gaussian Unitary Ensemble (GUE). More specifically, if n is the dimension of a random matrix from the GUE and k is such that both k and n-k tends to infinity as n tends to infinity then x_k is normally distributed in the limit. We also consider the joint limit distribution of x_k_1 < ... < x_k_m where we require that k_1, k_{i+1}-k_i and n-k_m, i=1..m-1, tends to infinity with n. The result is an m-dimensional Normal Distribution.

There is now a new version of the article on the Arxiv. Some minor errors have been corrected. One bad error was that the correlations in Theorem 1.4 were wrong 2007-04-04: I upload a new corrected version

Cited by in corpus (41)

• Edge Universality of Beta Ensembles
• Universality of Wigner random matrices: a Survey of Recent Results
• Random covariance matrices: Universality of local statistics of eigenvalues
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Universality of General $\beta$-Ensembles
• Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
• Large deviations of the maximal eigenvalue of random matrices
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
• Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble
• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Spectral measures of powers of random matrices
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Bulk properties of the Airy line ensemble
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Optimal local law and central limit theorem for $\beta$-ensembles
• Full counting statistics for interacting trapped fermions
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices
• Central limit theorem for partial linear eigenvalue statistics of Wigner matrices
• Some connections between the Classical Calogero-Moser model and the Log Gas
• Strong Convergence of Unitary Brownian Motion
• Local Central Limit Theorem for Determinantal Point Processes
• On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
• Rates of convergence for empirical spectral measures: a soft approach
• Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential
• Universal cutoff for Dyson Ornstein Uhlenbeck process
• Free energy fluctuations of the $2$-spin spherical SK model at critical temperature
• Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory
• Decomposition of spectral density in individual eigenvalue contributions
• On Eigenvalues of the sum of two random projections
• Uniform point variance bounds in classical beta ensembles
• Concentration of empirical distribution functions with applications to non-i.i.d. models
• Glass--like transition described by toppling of stability hierarchy
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Toeplitz determinants with a one-cut regular potential and Fisher--Hartwig singularities I. Equilibrium measure supported on the unit circle
• Crossover in densities of confined particles with finite range of interaction
• Global rigidity and exponential moments for soft and hard edge point processes
• On Fluctuations of Riemann's Zeta Zeros