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This article is dedicated to the following class of problems. Start with an $Ntimes N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1le t le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $tau=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dysons three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$.
For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initia
For the Langevin model of the dynamics of a Brownian particle with perturbations orthogonal to its current velocity, in a regime when the particle velocity modulus becomes constant, an equation for the characteristic function $psi (t,lambda )=Mleft[e
Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressio
With a scalar potential and a bivector potential, the vector field associated with the drift of a diffusion is decomposed into a generalized gradient field, a field perpendicular to the gradient, and a divergence-free field. We give such decompositio
We study the dynamics of the N-particle system evolving in the XY hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the