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About 10,000 papers for “Random Matrices and Applications”

PROPERTIES OF RANDOM MATRICES AND APPLICATIONS

Ian F. Blake, Chris Studholme · 2006

Abstract. This report surveys certain results on random matrices over finite fields and their applications, especially to coding theory. Extensive experimental work on such matrices is reported on and resulting conjectures are noted.

Cited by 0

Integrable systems, random matrices and applications

Dell'Atti, Marta · Northumbria Research Link (Northumbria University) · 2021

Nonlinear integrable systems emerge in a broad class of different problems in Mathematics and Physics.
\nOne of the most relevant characterisation of integrable systems is the existence of an infinite number of conservation laws, associated to integrable hierarchies of equations.
\nWhen nonlinearity is involved, critical phenomena may occur. A solution to a nonlinear partial differential equation may develop a gradient catastrophe and the consequent formation of a shock at the critical point. The approach of differential identities provides a convenient description of systems affected by phase transitions, identifying a suitable nonlinear equation for the order parameter of the system.
\nThis thesis is aimed to give a contribution to the perspective offered by the approach of differential identities. We discuss how this method is particularly useful in treating mean-field theories, with some explicit application. The core of the work concerns the Hermitian matrix ensemble and the symmetric matrix ensemble, analysed in the context of integrable systems. They both underlie a discrete integrable structure in form of a lattice, satisfying a discrete integrable hierarchy. We have studied a particular reduction of both system and determined the continuum limit of the dynamics of the field variables at the leading order.
\nParticular emphasis has been given to the study of the symmetric matrix ensemble. We have unveiled an unobserved double-chain structure shared by the field variables populating the lattice structure associated to the ensemble. In the continuum limit of a particular reduction of the lattice, we have found a new hydrodynamic chain, a hydrodynamic system with infinitely many components. We have shown that the hydrodynamic chain is integrable and we have conjectured the form of the associated hierarchy. The new integrable hydrodynamic chain constitutes per se an interesting object of study. Indeed, it presents some properties that are different from those shared by the standard integrable hydrodynamic chains studied in literature.

Cited by 0

Random matrices and applications to statistical signal processing

Pascal Vallet · theses.fr (ABES) · 2011

In this thesis, we consider the problem of source localization in large sensor networks, when the number of antennas of the network and the number of samples of the observed signal are large and of the same order of magnitude. We also consider the case where the source signals are deterministic, and we develop an improved algorithm for source localization, based on the MUSIC method. For this, we fist show new results concerning the position of the eigen values of large information plus noise complex gaussian random matrices

Cited by 0Open access

Random matrices and applications to statistical physics and quantum physics

Céline Nadal, MAJUMDAR, Satya · Munich Personal RePEc Archive (Ludwig Maximilian University of Munich) · 2011

Cette thèse est consacrée à l'étude des matrices aléatoires et à quelques unes de leurs applications en physique, en particulier en physique statistique et en physique quantique.C'est un travail essentiellement analytique complété par quelques simulations numériques Monte Carlo. Dans un premier temps j'introduis la théorie des matrices aléatoires de façon assez générale : je définis les principaux ensembles de matrices aléatoires (en particulier gaussiens) et décris leurs propriétés fondamentales (distribution des valeurs propres, densité, etc). Dans un second temps je m'intéresse à des systèmes physiques d'interfaces à l'équilibre qui peuvent être modélisés par des marcheurs vicieux'', c'est-à-dire des marcheurs aléatoires conditionnés à ne pas se croiser. On peut montrer que la distribution des positions des marcheurs à un temps donné est exactement celle des valeurs propres d'une matrice aléatoire. J'étudie ensuite un problème physique qui relève d'un domaine très différent, celui de l'information quantique, mais qui est également étroitement relié aux matrices aléatoires: celui de l'intrication pour des états aléatoires dans un système quantique bipartite (fait de deux sous-parties) de grande taille. Enfin je m'intéresse à certaines propriétés des matrices aléatoires comme la distribution du nombre de valeurs propres positives ou encore la distribution de la valeur propre maximale (loi de Tracy-Widom près de la moyenne et grandes déviations loin de la moyenne).

Cited by 0Open access

The Rank of Random Binary Matrices and Distributed Storage Applications

Paulo J. S. G. Ferreira, Bruno Bernardes de Jesus, José Vieira et al. · IEEE Communications Letters · 2012

Random binary matrices appear in a variety of signal processing and encoding problems. They play an important role in rateless codes and in distributed storage applications. This paper focuses on block angular matrices, a class of random rectangular binary matrices that are particularly suited to distributed storage applications. We address one of the key issues regarding binary random matrices in general, and block angular matrices in particular: the probability of obtaining a full rank matrix, when drawing uniformly at random from the set of binary matrices with compatible structure. This paper gives a closed-form expression for this probability, as well as some bounds and approximations.

Cited by 22

The matrix Dyson equation and its applications for random matrices

László Erdős · IAS/Park City mathematics series · 2019

These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau. We extend the scope of this book by focusing on new techniques developed to deal with generalizations of Wigner matrices that allow for non-identically distributed entries and even for correlated entries. This requires to analyze a system of nonlinear equations, or more generally a nonlinear matrix equation called the Matrix Dyson Equation (MDE). We demonstrate that stability properties of the MDE play a central role in random matrix theory. The analysis of MDE is based upon joint works with J. Alt, O. Ajanki, D. Schröder and T. Krüger that are supported by the ERC Advanced Grant, RANMAT 338804 of the European Research Council. The lecture notes were written for the 27th Annual PCMI Summer Session on Random Matrices held in 2017. The current edited version will appear in the IAS/Park City Mathematics Series, Vol. 26.

Cited by 10Open access

Topics in Random Matrices: Theory and Applications to Probability and Statistics

Termeh Kousha · uO Research (University of Ottawa) · 2011

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

Cited by 0Open access

Nonlinear Random Matrices and Applications to the Sum of Squares Hierarchy

Rajendran, Goutham · Knowledge@UChicago (University of Chicago) · 2022

We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved tremendous success for various problems in combinatorial optimization, robust statistics and machine learning. It's a family of convex relaxations that lets us smoothly trade off running time for approximation guarantees. In recent works, it's been shown to be extremely useful for recovering structure in high dimensional noisy data. It also remains our best approach towards refuting the notorious Unique Games Conjecture. In this work, we analyze the performance of the SoS hierarchy on fundamental problems stemming from statistics, theoretical computer science and statistical physics. In particular, we show subexponential-time SoS lower bounds for the problems of the Sherrington-Kirkpatrick Hamiltonian, Planted Slightly Denser Subgraph, Tensor Principal Components Analysis and Sparse Principal Components Analysis. These SoS lower bounds involve analyzing large random matrices, wherein lie our main contributions. These results offer strong evidence for the truth of and insight into the low-degree likelihood ratio hypothesis, an important conjecture that predicts the power of bounded-time algorithms for hypothesis testing. We also develop general-purpose tools for analyzing the behavior of random matrices which are functions of independent random variables. Towards this, we build on and generalize the matrix variant of the Efron-Stein inequalities. In particular, our general theorem on matrix concentration recovers various results that have appeared in the literature. We expect these random matrix theory ideas to have other significant applications.

Cited by 0Open access