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24
Random Matrices and the Riemann ZetaFunction: a Review
, 2003
"... The past few years have seen the emergence of compelling evidence for a connection between the zeros of the Riemann zeta function and the eigenvalues of random matrices. This hints at a link between the distribution of the prime numbers, which is governed by the Riemann zeros, and the properties of ..."
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The past few years have seen the emergence of compelling evidence for a connection between the zeros of the Riemann zeta function and the eigenvalues of random matrices. This hints at a link between the distribution of the prime numbers, which is governed by the Riemann zeros, and the properties of waves in complex systems (e.g. waves in random media, or in geometries where the ray dynamics is chaotic), which may be modelled using random matrix theory. These developments have led to a signicant deepening of our understanding of some of the most important problems relating to the zeta function and its kin, and have stimulated new avenues of research in random matrix theory. In particular, it would appear that several longstanding questions concerning the distribution of values taken by the zeta function on the line where the Riemann Hypothesis places its zeros can be answered using techniques developed in the study of random matrices. 1 Random matrices and the Riemann zeros Linear wave theories may expressed in terms of matrices. Therefore, just as in complex (e.g. chaotic) dynamical systems, where statistical properties of the trajectories may be calculated by averaging
Janossy densities for Unitary ensembles at the spectral edge
, 2009
"... For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in ..."
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For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated RiemannHilbert problem. 1
NUCLEI, PRIMES AND THE RANDOM MATRIX CONNECTION
, 2009
"... In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mat ..."
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In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics – random matrix theory – provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and
doi:10.1093/imrn/rnq019 ModPoisson Convergence in Probability and Number Theory
"... Building on earlier work introducing the notion of “modGaussian ” convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of “modPoisson ” convergence. We show in particular how it occurs naturally in analytic n ..."
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Building on earlier work introducing the notion of “modGaussian ” convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of “modPoisson ” convergence. We show in particular how it occurs naturally in analytic number theory in the classical Erdős– Kac Theorem. In fact, this case reveals deep connections and analogies with conjectures concerning the distribution of L functions on the critical line, which belong to the modGaussian framework, and with analogues over finite fields, where it can be seen as a zerodimensional version of the Katz–Sarnak philosophy in the “large conductor” limit. 1
An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles
, 2016
"... Abstract. We survey the current status of universality limits for mpoint correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed an ..."
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Abstract. We survey the current status of universality limits for mpoint correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider β ensembles for β = 2, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.
of a random permutation matrix
, 2000
"... random matrix theory, characteristic polynomial permutations, central limit theorem We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. With this result we can obtain a central limit theorem for the counting function for the eigenva ..."
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random matrix theory, characteristic polynomial permutations, central limit theorem We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. With this result we can obtain a central limit theorem for the counting function for the eigenvalues lying in some interval on the unit circle. 1
Research Statement
, 2009
"... My main interest is analytic number theory and random matrix theory (especially the distribution of zeros of ..."
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My main interest is analytic number theory and random matrix theory (especially the distribution of zeros of
Classical RMT Period m–Circulant Refs
, 2010
"... a11 a12 a13 · · · a1N a21 a22 a23 · · · a2N aN1 aN2 aN3 · · · aNN aij are functions of independent identically distributed random variables b1,...,bkN. Fix p, define Prob(A) = 1≤i≤kN p(bi). Example: Real symmetric ensemble. Pick entries of the matrix, up to equivalence of aij = aji, independen ..."
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a11 a12 a13 · · · a1N a21 a22 a23 · · · a2N aN1 aN2 aN3 · · · aNN aij are functions of independent identically distributed random variables b1,...,bkN. Fix p, define Prob(A) = 1≤i≤kN p(bi). Example: Real symmetric ensemble. Pick entries of the matrix, up to equivalence of aij = aji, independently from p. We have kN = N(N+1) 2 degrees of freedom.