Related papers: Lax random matrices from Calogero systems

High-dimensional settings, where the data dimension ($d$) far exceeds the number of observations ($n$), are common in many statistical and machine learning applications. Methods based on $\ell_1$-relaxation, such as Lasso, are very popular…

We develop a new, systematic approach towards studying the integrability of the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero models associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs…

The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S_{BGS} = - \int d{\bf H} [P({\bf H})] \ln [P({\bf H})], with suitable constraints. Here we construct and analyze…

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of $ad L$. For generic $L$ a…

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these…

The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important…

Using the Generalized Maximium Entropy Principle based on the nonextensive q entropy a new family of random matrix ensembles is generated. This family unifies previous extensions of Random Matrix Theory and gives rise to an orthogonal…

Matrix normal models have an associated 4-tensor for their covariance representation. The covariance array associated with a matrix normal model is naturally represented as a Kronecker-product structured covariance associated with the…

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the…

Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary…

We introduce two methods for estimating the density matrix for a quantum system: Quantum Maximum Likelihood and Quantum Variational Inference. In these methods, we construct a variational family to model the density matrix of a mixed…

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold…

We present a systematic study of the internal mass structure of early-type galaxies (ETGs) based on 106 galaxy-scale strong gravitational lenses with background quasars, all having spectroscopic redshifts. From this parent sample, we select…

We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a…

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of…

We study 2D non-linear sigma models on a group manifold with a special form of the metric. We address the question of integrability for this special class of sigma models. We derive two algebraic conditions for the metric on the group…

The structure of a polystyrene matrix filled with tightly cross-linked polystyrene nanoparticles, forming an athermal nanocomposite system, is investigated by means of a Monte Carlo sampling formalism. The polymer chains are represented as…

A family of integrable $GL(NM)$ models is described. On the one hand it generalizes the classical spin Ruijsenaars--Schneider systems (the case $N=1$), and on the other hand it generalizes the relativistic integrable tops on $GL(N)$ Lie…

Using molecular dynamics simulations, we study the slow dynamics of supercooled liquids confined in a random matrix of immobile obstacles. We study the dynamical crossover from glass-like to Lorentz-gas-like behavior in terms of the density…