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175
Random Matrix Theory and ζ(1/2 + it)
, 2000
"... We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the re ..."
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Cited by 157 (20 self)
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We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the
Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2001
"... This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian ..."
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Cited by 95 (15 self)
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This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.
Random matrices and determinantal processes
 Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005
"... Eigenvalues of random matrices have a rich mathematical structure and are a source of interesting distributions and processes. These distributions are natural statistical models in many problems in quantum physics, [15]. They occur for example, at least conjecturally, in the statistics of spectra of ..."
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Cited by 85 (5 self)
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Eigenvalues of random matrices have a rich mathematical structure and are a source of interesting distributions and processes. These distributions are natural statistical models in many problems in quantum physics, [15]. They occur for example, at least conjecturally, in the statistics of spectra of quantized models
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 62 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
The importance of Selberg integral
 Bull. Amer. Math. Soc
"... Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a que ..."
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Cited by 58 (8 self)
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Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of qanalogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero–Sutherland quantum manybody systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial
Mean values of Lfunctions and symmetry
 Int. Math. Res. Notices
"... Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollif ..."
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Cited by 54 (14 self)
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Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollified mean–square of the L–functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes–Vignéras Γ2–function and to a family of self–similar functions. 1.
Lowlying zeros of families of elliptic curves
, 2006
"... There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average di ..."
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Cited by 51 (3 self)
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There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of Lfunctions. We study the lowlying zeros for families of elliptic curve Lfunctions. For these Lfunctions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and SwinnertonDyer and the impressive partial results towards resolving the conjecture). We calculate the density of the lowlying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. A main challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great interest. An application is an improvement on the upper bound of the average rank of the family of all elliptic curves. We show that there is an extra contribution to the density of the lowlying zeros from the family with positive rank (presumably from the “extra ” zero at the central point). 1
On the frequency of vanishing of quadratic twists of modular Lfunctions
 in Number theory for the millennium, I (Urbana, IL, 2000), 301–315, A K Peters
, 2002
"... Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishi ..."
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Cited by 49 (16 self)
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Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular Lfunctions. The central question is the following: given a holomorphic newform f with integral coefficients and associated Lfunction Lf(s), for how many fundamental discriminants d with d  ≤ x, does Lf(s, χd), the Lfunction twisted by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the Lfunction is associated with an elliptic curve, in light of the conjecture of Birch and SwinnertonDyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated Lfunction (1) LE(s) = for ℜs> 1. Then, as a consequence of the TaniyamaShimura conjecture, recently solved by Wiles, Taylor, ([W], [TW]), and Breuil, Conrad, and Diamond, LE is entire and satisfies a functional equation n=1 a ∗ n n s