β-Hermite Ensemble Research Articles

Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials

N-dimensional Bessel and Jacobi processes describe interacting particle systems with N particles and are related to β-Hermite, β-Laguerre, and β-Jacobi ensembles. For fixed N, there exist associated weak limit theorems (WLTs) in the freezing regime β → ∞ in the β-Hermite and β-Laguerre case by Dumitriu and Edelman [Ann. Inst. Henri Poincare, Sect. B 41, 1083 (2005)] with explicit formulas for the covariance matrices ΣN in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed ΣN−1 with formulas for the eigenvalues and eigenvectors of ΣN−1 and thus of ΣN. In the present paper, we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for ΣN from ΣN−1, where, for β-Hermite and β-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for N → ∞ in terms of the Airy function. For β-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.

Lower deviations in β-ensembles and law of iterated logarithm in last passage percolation

For last passage percolation (LPP) on ℤ2 with exponential passage times, let Tn denote the passage time from (1, 1) to (n,n). We investigate the law of iterated logarithm of the sequence {Tn}n≥1; we show that $$\lim \,{\inf _{n \to \infty }}{{{T_n} - 4n} \over {{n^{1/3}}{{\left( {\log \,\log \,n} \right)}^{1/3}}}}$$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight shift in perspective from point-to-point passage times to considering point-to-line passage times instead, and exploiting the correspondence of the latter to the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient, which is of independent interest, is a new lower bound of lower tail deviation probability of the largest eigenvalue of β-Laguerre ensembles, which extends the results proved in the context of the β-Hermite ensembles by Ledoux and Rider (2010).

Crossover in nonstandard random-matrix spectral fluctuations without unfolding.

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the β-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.

The Tracy–Widom distribution is not infinitely divisible

The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the β-Tracy–Widom distribution, which is the limiting distribution of the largest eigenvalue of a β-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed N≥2 it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.

A pseudo-Hermitian [formula omitted]-Hermite family of matrices

A family of pseudo-Hermitian random matrices is constructed whose eigenvalues are the same of the β-Hermite ensemble of random matrix theory. It is shown that matrices of this family are unstable in such a way that small perturbations cause the eigenvalues to move into the complex plane in conjugate pairs.

A result on precise asymptotics for largest eigenvalues of β ensembles

The paper focuses on the precise asymptotics of the largest eigenvalues of β-Hermite ensemble and β-Laguerre ensemble. In particular, we obtain a general law on the precise moment convergence rates for a type of weighted series constructed by the first-order conditional moment of the largest eigenvalues of β ensembles. The results are motivated by the complete convergence for partial sums of independent random variables. The proofs depend on the small deviations for largest eigenvalue of β ensembles and non-asymptotic tail probability inequalities of the general β Tracy-Widom law.

The moment convergence rates for largest eigenvalues of β ensembles

The paper focuses on the largest eigenvalues of the β-Hermite ensemble and the β-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d. random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the general β Tracy-Widom law.

Fixed trace β-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density

In the present paper, fixed trace β-Hermite ensembles generalizing the fixed trace Gaussian ensembles are considered. For all β, we prove the Wigner semicircle law for these ensembles by using two different methods: one is the moment equivalence method with the help of the matrix model for general β, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for β-HE implies the same limit for fixed trace β-Hermite ensembles. Consequently, explicit limit can be given for fixed trace Gaussian orthogonal, unitary, and symplectic ensembles. Furthermore, for even β, analogous to β-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.

Sturm Sequences and Random Eigenvalue Distributions

This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the β-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n 2+m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of this approach, we give a derivation of both distributions for the β-Hermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the β-Hermite matrix ensemble.

Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

The evolution with β of the distributions of the spacing ‘ s’ between nearest-neighbour levels of unfolded spectra of random matrices from the β-Hermite ensemble ( β-HE) is investigated by Monte Carlo simulations. The random matrices from the β-HE are real symmetric and tridiagonal where β, which can take any positive value, is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues. The distribution of eigenvalues coincide with those of the three classical Gaussian ensembles for β=1, 2, 4. The use of the β-HE ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices. Generalized gamma distributions are shown to be excellent approximations of the nearest-neighbor spacing (NNS) distributions for any β while being still simple. They account both for the level repulsion in ∼ s β when s→0 and for the whole shape of the NNS distributions in the range of ‘ s’ which is accessible to experiment or to most numerical simulations. The exact NNS distribution of the GOE ( β=1) is in particular significantly better described by a generalized gamma distribution than it is by the Wigner surmise while the best generalized gamma approximation coincides essentially with the Wigner surmise for β>∼2. They describe too the evolution of the level repulsion between that of a Poisson distribution and that of a GOE distribution when β increases from 0 to 1. The distribution of ln ( s), related to the electrostatic interaction energy between neighbouring charges, is accordingly well approximated by a generalized Gumbel distribution for any β⩾0. The distributions of the minimum NN spacing between eigenvalues of matrices from the β-HE, obtained both from as-calculated eigenvalues and from unfolded eigenvalues are Brody distributions which are classically used to characterize the spectral fluctuations of various physical systems.

The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N × N β-Hermite matrix

The β-Hermite ensemble (β-HE) of tridiagonal N × N random matrices of Dumitriu and Edelman (2002 J. Math. Phys. 43 5830) is a continuum of ensembles in which β, the reciprocal of the temperature in the 2D electrostatic interpretation of the eigenvalue characteristics, can take any value. The eigenvalue distributions coincide with those of the classical Gaussian ensembles (GOE, GUE, GSE) for β = 1, 2, 4. A fixed-trace β-Hermite ensemble (β-FTHE) is defined from the β-HE and is used to extend the spherical ensembles of classical symmetries to β-spherical ensembles. At low temperature, when β → ∞, for a fixed value of N, the asymptotic distributions of reduced determinants DN,β of random N × N β-H and β-FTH matrices are shown to be standard Gaussians. Accordingly, the fluctuations of the potential at the origin, −ln |DN,β|, have a generalized Gumbel distribution at low temperature. For large N and large β, a ln(N) variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions.

A matrix model for the β-Jacobi ensemble

This note presents a random matrix model for general (β>0) β-Jacobi ensembles. This generalizes the well-known MANOVA models for β=1,2,4 and eliminates the quantization of β (and other parameters) present in the previously known models. This model is a partial answer to an open problem presented by Dumitriu and Edelman, where they also presented models for the β-Laguerre and β-Hermite ensembles.