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The behavior of a stationary inverted point mass pendulum pivoted at its lower end in a gravitational potential is studied under the influence of statistical fluctuations. It is shown using purely classical equations that the pendulum…
We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the…
We present a new, short, self-contained proof of localization properties of multi-dimensional continuum random Schr\"odinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment…
The classical limit of the Wigner-Weyl representation is used to approximate products of bound-continuum matrix elements that are fundamental to many coherent control computations. The range of utility of the method is quantified through an…
New time dependent Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been presented as the wave equation of a 2D membrane in the phase plane. The values…
We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and…
As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral…
We present a semiclassical calculation of the generalized form factor which characterizes the fluctuations of matrix elements of the quantum operators in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some…
We derive a simple formula for the fluctuations of the time average around the thermal mean for overdamped Brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, et al. in the context of…
Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem for moments spectral measure is also derived. The proof is…
We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We find that the fluctuation of this difference is much smaller than those of the…
We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of…
Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon…
In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which…
The joint moments of the derivatives of the characteristic polynomial of a random unitary matrix, and also a variant of the characteristic polynomial that is real on the unit circle, in the large matrix size limit, have been studied…
We prove a moment majorization principle for matrix-valued functions with domain $\{-1,1\}^{m}$, $m\in\mathbb{N}$. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random…
Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $\mu_1(M_n), \mu_2(M_n), ..., \mu_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$…
Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and…
A recently proposed Boltzmann local equilibrium Wigner function for massive spin-1/2 particles is generalized to the case of Fermi-Dirac statistics. The resulting formula ensures the correct normalization of the mean polarization vector and…
Given standard angular momentum and boost matrices, the commutation rules for vector and momentum matrices are solved. The resulting matrix components are displayed as detailed functions of spin with factors such as the square root of…