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Madan Lal Mehta

Bio: Madan Lal Mehta is an academic researcher from University of Delhi. The author has contributed to research in topics: Matrix (mathematics) & Random matrix. The author has an hindex of 14, co-authored 25 publications receiving 2399 citations. Previous affiliations of Madan Lal Mehta include Centre national de la recherche scientifique & Jawaharlal Nehru University.

Papers
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Journal ArticleDOI
TL;DR: In this article, the integral over twoon ×n hermitan matrices Z(g, c) = ∫dAdBexp{−tr[A¯¯¯¯2+B��2−2cAB+g/n(A¯¯4+B��4)]} is evaluated in the limit of largen.
Abstract: The integral over twon ×n hermitan matricesZ(g, c)=∫dAdBexp{−tr[A 2+B 2−2cAB+g/n(A 4+B 4)]} is evaluated in the limit of largen. For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight. The above integral arises in the study of the planar approximation to quantum field theory.

318 citations

Journal ArticleDOI
TL;DR: In this article, the general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant, using a slight generalization of a theorem of Dyson.
Abstract: The general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.

252 citations

Journal ArticleDOI
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.

224 citations


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Book
01 Jan 2004
TL;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 2004
TL;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

Journal ArticleDOI
TL;DR: In this paper, the authors derived analogues for the Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E., the expression of the Fredholm determinant in terms of a Painleve transcendent, the existence of a commuting differential operator, and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.
Abstract: Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

1,923 citations

Journal ArticleDOI
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

Book
21 Dec 2009
TL;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