On the density of eigenvalues of a random matrix
TL;DR: An exact expression for the density of eigenvalues of a random-matrix is derived and it can be seen very directly that it goes over to Wigner's “semi-circle law” when the order of the matrix becomes infinite.
About: This article is published in Nuclear Physics.The article was published on 1960-08-01. It has received 224 citations till now. The article focuses on the topics: Matrix differential equation & Random matrix.
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01 Jan 2004TL;DR: In this paper, the critical zeros of the Riemann zeta function are defined and the spacing of zeros is defined. But they are not considered in this paper.
Abstract: Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.
3,399 citations
Book•
28 Jun 2004TL;DR: A tutorial on random matrices is provided which provides an overview of the theory and brings together in one source the most significant results recently obtained.
Abstract: Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.
2,308 citations
TL;DR: In this article, three kinds of statistical ensembles are defined, representing a mathematical idealization of the notion of ''all physical systems with equal probability'' and three groups are studied in detail, based mathematically upon the orthogonal, unitary and symplectic groups.
Abstract: New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.
1,913 citations
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21 Dec 2009TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.
1,289 citations
Book•
13 Apr 2012TL;DR: The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics as mentioned in this paper, which makes the current state of the field almost too large to survey in a single book.
Abstract: The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.
1,075 citations
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TL;DR: In this paper, the level-spacing distribution under the random-matrix hypothesis is studied, and lower and upper bounds for the distribution function of the levelspacing are given.
165 citations
148 citations
TL;DR: In this article, the distribution of the largest, smallest and any intermediate root when the roots are specified by their position in a monotonic arrangement has been derived for $p = 2, 3, 4,$ and 5 by the new method.
Abstract: S. N. Roy [2] obtained in 1943 the distribution of the maximum, minimum and any intermediate one of the roots of certain determinantal equations based on covariance matrices of two samples on the null hypothesis of equal covariance matrices in the two populations. The present paper gives a different method of working out the distribution of any of these roots under the same hypothesis. The distribution of the largest, smallest and any intermediate root when the roots are specified by their position in a monotonic arrangement has been derived for $p = 2, 3, 4,$ and 5 by the new method. The method is applicable for obtaining the distribution of the roots of an equation of any order, when the distributions of the roots of lower order equations have been worked out.
30 citations