Random Matrices, Frobenius Eigenvalues, and Monodromy (original) (raw)

Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory

Geometric and Functional Analysis, 2013

The statistics of low-lying zeros of quadratic Dirichlet Lfunctions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao's combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz-Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g → ∞ allows us to identify Gao's combinatorial factor with the RMT answer.

Random matrices and Riemann hypothesis

2011

The curious connection between the spacings of the eigenvalues of random matrices and the corresponding spacings of the nontrivial zeros of the Riemann zeta function is analyzed on the basis of the geometric dynamical global program of Langlands whose fundamental structures are shifted quantized conjugacy class representatives of bilinear algebraic semigroups. The considered symmetry behind this phenomenology is the differential bilinear Galois semigroup shifting the product, right by left, of automorphism semigroups of cofunctions and functions on compact transcendental quanta.

Random matrix theory over finite fields

Bulletin of The American Mathematical Society, 2001

The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various measures on partitions.

Large spaces between the zeros of the Riemann zeta-function and random matrix theory

Journal of Number Theory, 2004

On the hypothesis that the 2k-th mixed moments of Hardy's Z-function and its derivative are correctly predicted by random matrix theory, it is established that large gaps (depending on, and apparently increasing with k) exist between the zeta zeros. The case k ¼ 3 has been worked out in an earlier paper (in this journal) and the cases k ¼ 4; 5; 6 are considered here. When k ¼ 6 the gaps obtained have 44 times the average gap length. This depends on calculations involving Jacobi-Schur functions and formulae for these functions due to Jacobi, Trudi and Aitken in the classical theory of equations.

Combinatorial theory of permutation-invariant random matrices III: random walks on S(N), ramified coverings and the large N S(N)-Yang-Mills measure

The aim of this article is to study some asymptotics of a natural model of random ramified coverings on the disk of degree N. We prove that the monodromy field, called also the holonomy field, converges in probability to a non-random field as N goes to infinity. In order to do so, we use the fact that the monodromy field of random uniform labelled simple ramified coverings on the disk of degree N has the same law as the S(N)-Yang-Mills measure associated with the random walk by transposition on S(N). This allows us to restrict our study to random walks on S(N): we prove theorems about asymptotic of random walks on S(N) in a new framework based on the geometric study of partitions and the Schur-Weyl-Jones’s dualities. In particular, given a sequence of conjugacy classes, we define a notion of convergence for this sequence which implies the convergence in non-commutative distribution and in P-expectation of the random walk to a P-free multiplicative Lévy process. This limiting process is shown not to be a free multiplicative Lévy process and we compute its log-cumulant functional. We give also a criterion on the sequence of conjugacy classes in order to know if the limit is random or not.

On the Moments of Traces of Matrices of Classical Groups

Communications in Mathematical Physics, 2004

We consider random matrices, belonging to the groups U (n), O(n), SO(n), and Sp(n) and distributed according to the corresponding unit Haar measure. We prove that the moments of traces of powers of the matrices coincide with the moments of certain Gaussian random variables if the order of moments is low enough. Corresponding formulas, proved partly before by various methods, are obtained here in the framework of a unique method, reminiscent of the method of correlation equations of statistical mechanics. The equations are derived by using a version of the integration by parts.

A Note on the Eigenvalue Density of Random Matrices

Communications in Mathematical Physics, 1999

The distribution of eigenvalues of N × N random matrices in the limit N → ∞ is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.

Random Matrix Theory and the Fourier Coefficients of Half-Integral-Weight Forms

Experimental Mathematics, 2006

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral weight modular forms. Numerical evidence is presented in support of them.

The Schur Expansion of Characteristic Polynomials and Random Matrices

2021

We develop a new framework to compute the exact correlators of characteristic polynomials, and their inverses, in random matrix theory. Our results hold for general potentials and incorporate the effects of an external source. In matrix model realizations of string theory, these correspond to correlation functions of exponentiated “(anti-)branes” in a given background of “momentum branes”. Our method relies on expanding the (inverse) determinants in terms of Schur polynomials, then re-summing their expectation values over the allowed representations of the symmetric group. Beyond unifying previous, seemingly disparate calculations, this powerful technique immediately delivers two new results: 1) the full finite N answer for the correlator of inverse determinant insertions in the presence of a matrix source, and 2) access to an interesting, novel regime M > N , where the number of inverse determinant insertions M exceeds the size of the matrix N . taro.kimura@u-bourgogne.fr mazenc...