A random matrix decimation procedure relating β=2/(r+1)\beta = 2/(r+1) to β=2(r+1)\beta = 2(r+1)

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another $\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur

Scaling of Level Statistics at the Disorder-Induced Metal-Insulator Transition

The distribution of energy level separations for lattices of sizes up to 28$\times$28$\times$28 sites is numerically calculated for the Anderson model. The results show one-parameter scaling. The size-independent universality of the critical level spacing distribution allows to detect with high precision the critical disorder $W_{c}=16.35$. The scaling properties yield the critical exponent, $\nu =1.45 \pm 0.08$, and the disorder dependence of the correlation length.Comment: 11 pages (RevTex), 3 figures included (tar-compressed and uuencoded using UUFILES), to appear in Phys.Rev. B 51 (Rapid Commun.

Anderson transitions in three-dimensional disordered systems with randomly varying magnetic flux

The Anderson transition in three dimensions in a randomly varying magnetic flux is investigated in detail by means of the transfer matrix method with high accuracy. Both, systems with and without an additional random scalar potential are considered. We find a critical exponent of $\nu=1.45\pm0.09$ with random scalar potential. Without it, $\nu$ is smaller but increases with the system size and extrapolates within the error bars to a value close to the above. The present results support the conventional classification of universality classes due to symmetry.Comment: 4 pages, 2 figures, to appear in Phys. Rev.

Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.Comment: 28 page

Eigenvector localization for random band matrices with power law band width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width $W$ raised to a power $\mu$ remains smaller than the matrix size $N$. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width $W$, the estimate $\mu \le 8$ holds.Comment: 30 pages; corrected typo

Spectral correlations : understanding oscillatory contributions

We give a different derivation of a relation obtained using a supersymmetric nonlinear sigma model by Andreev and Altshuler [Phys. Rev. Lett. 72, 902 (1995)], which connects smooth and oscillatory components of spectral correlation functions. We show that their result is not specific to the random matrix theory. Also, we show that despite an apparent contradiction, the results obtained using their formula are consistent with earlier perspectives on random matrix models

Universal Correlations of Coulomb Blockade Conductance Peaks and the Rotation Scaling in Quantum Dots

We show that the parametric correlations of the conductance peak amplitudes of a chaotic or weakly disordered quantum dot in the Coulomb blockade regime become universal upon an appropriate scaling of the parameter. We compute the universal forms of this correlator for both cases of conserved and broken time reversal symmetry. For a symmetric dot the correlator is independent of the details in each lead such as the number of channels and their correlation. We derive a new scaling, which we call the rotation scaling, that can be computed directly from the dot's eigenfunction rotation rate or alternatively from the conductance peak heights, and therefore does not require knowledge of the spectrum of the dot. The relation of the rotation scaling to the level velocity scaling is discussed. The exact analytic form of the conductance peak correlator is derived at short distances. We also calculate the universal distributions of the average level width velocity for various values of the scaled parameter. The universality is illustrated in an Anderson model of a disordered dot.Comment: 35 pages, RevTex, 6 Postscript figure

Scaling Analysis of Fluctuating Strength Function

We propose a new method to analyze fluctuations in the strength function phenomena in highly excited nuclei. Extending the method of multifractal analysis to the cases where the strength fluctuations do not obey power scaling laws, we introduce a new measure of fluctuation, called the local scaling dimension, which characterizes scaling behavior of the strength fluctuation as a function of energy bin width subdividing the strength function. We discuss properties of the new measure by applying it to a model system which simulates the doorway damping mechanism of giant resonances. It is found that the local scaling dimension characterizes well fluctuations and their energy scales of fine structures in the strength function associated with the damped collective motions.Comment: 22 pages with 9 figures; submitted to Phys. Rev.

The Ground State Energy of Dilute Bose Gas in Potentials with Positive Scattering Length

The leading term of the ground state energy/particle of a dilute gas of bosons with mass $m$ in the thermodynamic limit is $2\pi \hbar^2 a \rho/m$ when the density of the gas is $\rho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a>0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.Comment: Latex 28 page

Dyson processes on the octonion algebra

We consider Brownian motion on symmetric matrices of octonions, and study the law of the spectrum. Due to the fact that the octonion algebra is nonassociative, the dimension of the matrices plays a special role. We provide two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent in the law of the spectrum