7,474 research outputs found
Singularity dominated strong fluctuations for some random matrix averages
The circular and Jacobi ensembles of random matrices have their eigenvalue
support on the unit circle of the complex plane and the interval of the
real line respectively. The averaged value of the modulus of the corresponding
characteristic polynomial raised to the power diverges, for , at points approaching the eigenvalue support. Using the theory of
generalized hypergeometric functions based on Jack polynomials, the functional
form of the leading asymptotic behaviour is established rigorously. In the
circular ensemble case this confirms a conjecture of Berry and Keating.Comment: 11 pages, to appear Commun. Math. Phy
Nodal domain distributions for quantum maps
The statistics of the nodal lines and nodal domains of the eigenfunctions of
quantum billiards have recently been observed to be fingerprints of the
chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett.,
Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88
(2002), 114102). These statistics were shown to be computable from the random
wave model of the eigenfunctions. We here study the analogous problem for
chaotic maps whose phase space is the two-torus. We show that the distributions
of the numbers of nodal points and nodal domains of the eigenvectors of the
corresponding quantum maps can be computed straightforwardly and exactly using
random matrix theory. We compare the predictions with the results of numerical
computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction
Moments of zeta and correlations of divisor-sums: III
In this series we examine the calculation of the th moment and shifted
moments of the Riemann zeta-function on the critical line using long Dirichlet
polynomials and divisor correlations. The present paper is concerned with the
precise input of the conjectural formula for the classical shifted convolution
problem for divisor sums so as to obtain all of the lower order terms in the
asymptotic formula for the mean square along of a Dirichlet polynomial
of length up to with divisor functions as coefficients
A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros
We illustrate a general method for calculating spectral statistics that
combines the universal (Random Matrix Theory limit) and the non-universal
(trace-formula-related) contributions by giving a heuristic derivation of the
three-point correlation function for the zeros of the Riemann zeta function.
The main idea is to construct a generalized Hermitian random matrix ensemble
whose mean eigenvalue density coincides with a large but finite portion of the
actual density of the spectrum or the Riemann zeros. Averaging the random
matrix result over remaining oscillatory terms related, in the case of the zeta
function, to small primes leads to a formula for the three-point correlation
function that is in agreement with results from other heuristic methods. This
provides support for these different methods. The advantage of the approach we
set out here is that it incorporates the determinental structure of the Random
Matrix limit.Comment: 22 page
Two-point correlation function for Dirichlet L-functions
The two-point correlation function for the zeros of Dirichlet L-functions at
a height E on the critical line is calculated heuristically using a
generalization of the Hardy-Littlewood conjecture for pairs of primes in
arithmetic progression. The result matches the conjectured Random-Matrix form
in the limit as and, importantly, includes finite-E
corrections. These finite-E corrections differ from those in the case of the
Riemann zeta-function, obtained in (1996 Phys. Rev. Lett. 77 1472), by certain
finite products of primes which divide the modulus of the primitive character
used to construct the L-function in question.Comment: 10 page
Resummation and the semiclassical theory of spectral statistics
We address the question as to why, in the semiclassical limit, classically
chaotic systems generically exhibit universal quantum spectral statistics
coincident with those of Random Matrix Theory. To do so, we use a semiclassical
resummation formalism that explicitly preserves the unitarity of the quantum
time evolution by incorporating duality relations between short and long
classical orbits. This allows us to obtain both the non-oscillatory and the
oscillatory contributions to spectral correlation functions within a unified
framework, thus overcoming a significant problem in previous approaches. In
addition, our results extend beyond the universal regime to describe the
system-specific approach to the semiclassical limit.Comment: 10 pages, no figure
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