Random matrix ensembles with an effective extensive external charge

Abstract

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta =2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.

LaTeX209, 17 pages

References in corpus (4)

• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Quantum mechanical time-delay matrix in chaotic scattering
• Novel universal correlations in invariant random-matrix models
• Universality in invariant random-matrix models: Existence near the soft edge

Cited by in corpus (16)

• Noise Dressing of Financial Correlation Matrices
• Induced measures in the space of mixed quantum states
• Random Matrix Theory Analysis of Cross Correlations in Financial Markets
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Distribution of the quantum mechanical time-delay matrix for a chaotic cavity
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• Moments of the transmission eigenvalues, proper delay times and random matrix theory II
• Random Matrix Theory and Cross-correlations in Global Financial Indices and Local Stock Market Indices
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• The difference between two random mixed quantum states: exact and asymptotic spectral analysis
• Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-ensembles
• Non-interacting fermions in hard-edge potentials
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• Delay-Time and Thermopower Distributions at the Spectrum Edges of a Chaotic Scatterer
• Characterizing price index behavior through fluctuation dynamics