10,207 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
Twin prime correlations from the pair correlation of Riemann zeros
We establish, via a formal/heuristic Fourier inversion calculation, that the
Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula
for the two-point correlation function of Riemann zeros at a height on the
critical line. Previously it was known that the Hardy-Littlewood conjecture
implies the pair correlation formula, and we show that the reverse implication
also holds. A smooth form of the Hardy-Littlewood conjecture is obtained by
inverting the limit of the two-point correlation
function and the precise form of the conjecture is found by including
asymptotically lower order terms in the two-point correlation function formula.Comment: 11 page
Conjectures for the integral moments and ratios of L-functions over function fields
We extend to the function field setting the heuristic previously developed,
by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and
ratios of -functions defined over number fields. Specifically, we give a
heuristic for the moments and ratios of a family of -functions associated
with hyperelliptic curves of genus over a fixed finite field
in the limit as . Like in the number field
case, there is a striking resemblance to the corresponding formulae for the
characteristic polynomials of random matrices. As an application, we calculate
the one-level density for the zeros of these -functions.Comment: 40 page
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
Value distribution of the eigenfunctions and spectral determinants of quantum star graphs
We compute the value distributions of the eigenfunctions and spectral
determinant of the Schrodinger operator on families of star graphs. The values
of the spectral determinant are shown to have a Cauchy distribution with
respect both to averages over bond lengths in the limit as the wavenumber tends
to infinity and to averages over wavenumber when the bond lengths are fixed and
not rationally related. This is in contrast to the spectral determinants of
random matrices, for which the logarithm is known to satisfy a Gaussian limit
distribution. The value distribution of the eigenfunctions also differs from
the corresponding random matrix result. We argue that the value distributions
of the spectral determinant and of the eigenfunctions should coincide with
those of Seba-type billiards.Comment: 32 pages, 9 figures. Final version incorporating referee's comments.
Typos corrected, appendix adde
On relations between one-dimensional quantum and two-dimensional classical spin systems
We exploit mappings between quantum and classical systems in order to obtain
a class of two-dimensional classical systems with critical properties
equivalent to those of the class of one-dimensional quantum systems discussed
in a companion paper (J. Hutchinson, J. P. Keating, and F. Mezzadri,
arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki
mapping; the method of coherent states; and a calculation based on commuting
the quantum Hamiltonian with the transfer matrix of a classical system. This
enables us to establish universality of certain critical phenomena by extension
from the results in our previous article for the classical systems identified.Comment: 36 page
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
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