Finite-N fluctuation formulas for random matrices

Abstract

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic N j=1 x j − x is computed exactly and shown to satisfy a central limit theorem as N → ∞. For the circular random matrix ensemble the p.d.f.'s for the linear statistics 1 2 N j=1 (θ j −π) and − N j=1 log 2| sin θ j /2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as N → ∞.

Coulomb Blockade Peak Spacing Fluctuations in Deformable Quantum Dots: A Further Test of Random Matrix Theory

“…We remark that some analytical work [23], as well as the Dyson Brownian motion model [2] hint the idea that the position of each level taken alone is Gaussian distributed, although we found no rigorous proof in the literature. So far we have not considered the possibility of having spin degenerate eigenstates.…”

Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

“…In particular the Selberg correlation integrals were evaluated in terms of a generalization of the Gauss hypergeometric function 2 F 1 based on the Jack polynomials. It was realized by one of the present authors [10,11,12,13,14,15] that theory initiated in [8] could be further developed and used to express the average of (9) with respect to (14) and its limiting forms as the Laguerre -(2λ) ensemble and the Gaussian -(2λ) ensemble, as m -dimensional integrals. Because the role of n and m is effectively interchanged, these integration identities have been referred to as duality formulas [16,17,18].…”

Definitions and Basic Properties

“…We obtain the exact distribution of sufficient statistics for the four classical β-ensembles: β-Hermite, β-Laguerre, β-Jacobi and Cauchy unitary ensemble. Exact distributions of certain linear statistics for β-Hermite and β-Laguerre ensemble have been computed by Baker and Forrester [18]. We reestablish and extend some of their results.…”

“…In sections 4, 5, 6, 7 we derive exact probability distributions for certain linear statistics arising in the study of random matrix ensembles. In particular, we reestablish and extend some of the results of Baker, Forrester [18]. These results are further incorporated into the maximum likelihood estimation problem.…”

“…. , θ r ) as the r-dimensional parameter, C n (θ) as the normalizing constant and 18) where real-valued functions V j with Fourier coefficients {V j (k)} k∈Z are assumed to satisfy the condition…”