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A statistical method for the estimation of spatial correlation lengths of Gaussian-shaped fluctuating signals with two measurement points is examined to quantitatively evaluate its reliability (variance) and accuracy (bias error). The…
An approach is suggested for treating multiscale fluctuations in macromolecular systems. The emphasis is on the statistical properties of such fluctuations. The approach is illustrated by a macromolecular system with mesoscopic fluctuations…
We show that correlation matrices with particular average and variance of the correlation coefficients have a notably restricted spectral structure. Applying geometric methods, we derive lower bounds for the largest eigenvalue and the…
This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrix. Under quite general assumptions, we prove that the traces are approximately normal distributed. Multi-dimensional central limit…
We consider a system of diffusing particles on the real line in a quadratic external potential and with repulsive electrostatic interaction. The empirical measure process is known to converge weakly to a deterministic measure-valued process…
Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas of inelastic hard spheres near the threshold of the clustering instability. They are shown to be governed by vorticity fluctuations only,…
We study mesoscopic fluctuations of the conductance through a quantum dot at the wings of the Coulomb blokade peaks. At low temperatures, the main mechanism of conduction is the elastic cotunneling. The conductance strongly fluctuates with…
We consider the asymptotic behavior of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized…
We study the statistics of a system of N random levels with integer values, in the presence of a logarithmic repulsive potential of Dyson type. This probleme arises in sums over representations (Young tableaux) of GL(N) in various matrix…
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…
For $k,m,n\in \mathbb{N}$, we consider $n^k\times n^k$ random matrices of the form $$ \mathcal{M}_{n,m,k}(\mathbf{y})=\sum_{\alpha=1}^m\tau_\alpha {Y_\alpha}Y_\alpha^T,\quad…
Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To…
The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering and social sciences as a tool to understand the information contained in typically large datasets. In…
The spectral fluctuations of complex quantum systems, in appropriate limit, are known to be consistent with that obtained from random matrices. However, this relation between the spectral fluctuations of physical systems and random matrices…
We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $\frac{d}{2}$, where $d$…
We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases when the correlation length is finite, the law of…
We study the statistical fluctuations (such as the variance) of causal set quantities, with particular focus on the causal set action. To facilitate calculating such fluctuations, we develop tools to account for correlations between causal…
The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such…
We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…
This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in…