Related papers: Novel universal correlations in invariant random-m…
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…
Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties…
We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra…
The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex…
We study invariant random matrix ensembles \begin{equation*} \mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the…
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical…
In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives…
In the past 20 years, the study of real eigenvalues of non-symmetric real random matrices has seen important progress. Notwithstanding, central questions still remain open, such as the characterization of their asymptotic statistics and the…
The Novikov-Shubin invariant associated to a graph provides information about the accumulation of eigenvalues of the corresponding adjacency matrix close to the origin. For a directed graph these eigenvalues lie in the complex plane and…
The nuclear equation of state, which determines the structure and properties of neutron stars, remains subject to substantial theoretical uncertainties, leading to model dependence in predicted observables. Universal relations have emerged…
The complicated interactions in presence of disorder lead to a correlated randomization of states. The Hamiltonian as a result behaves like a multi-parametric random matrix with correlated elements. We show that the eigenvalue correlations…
The ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential are studied. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several…
Motivated by the phenomenologies of dynamic roughening of strings in random media and magnetohydrodynamics, we examine the universal properties of driven diffusive system with coupled fields. We demonstrate that cross-correlations between…
The smoothed correlation function for the eigenvalues of large hermitian matrices, derived recently by Brezin and Zee [Nucl. Phys. B402 (1993) 613], is generalized to all random-matrix ensembles of Wigner-Dyson type. Submitted to Nuclear…
We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths $\lambda =1/ 2$, $1$,…
{Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous…
We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose $r$-th absolute moment decays as $N^{-1-(r-2)\epsilon}$ for some $\epsilon>0$ are universal. This includes sparse matrices…
Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of…
We prove the universality of the large deviations for conjugacy invariant permutations with few cycles. As an application, we establish the universality of large deviation at speeds $n$ and $\sqrt{n}$ for the length of monotone subsequences…
The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results…