176 research outputs found
Random matrix theory, the exceptional Lie groups, and L-functions
There has recently been interest in relating properties of matrices drawn at
random from the classical compact groups to statistical characteristics of
number-theoretical L-functions. One example is the relationship conjectured to
hold between the value distributions of the characteristic polynomials of such
matrices and value distributions within families of L-functions. These
connections are here extended to non-classical groups. We focus on an explicit
example: the exceptional Lie group G_2. The value distributions for
characteristic polynomials associated with the 7- and 14-dimensional
representations of G_2, defined with respect to the uniform invariant (Haar)
measure, are calculated using two of the Macdonald constant term identities. A
one parameter family of L-functions over a finite field is described whose
value distribution in the limit as the size of the finite field grows is
related to that of the characteristic polynomials associated with the
7-dimensional representation of G_2. The random matrix calculations extend to
all exceptional Lie groupsComment: 14 page
Weighted Bergman kernels and virtual Bergman kernels
We introduce the notion of "virtual Bergman kernel" and apply it to the
computation of the Bergman kernel of "domains inflated by Hermitian balls", in
particular when the base domain is a bounded symmetric domain.Comment: 12 pages. One-hour lecture for graduate students, SCV 2004, August
2004, Beijing, P.R. China. V2: typo correcte
Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
Fully chaotic Hamiltonian systems possess an infinite number of classical
solutions which are periodic, e.g. a trajectory ``p'' returns to its initial
conditions after some fixed time tau_p. Our aim is to investigate the spectrum
tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for
the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the
eigenvalues of the classical evolution operator. The density is naturally
decomposed into a smooth part plus an interferent sum over oscillatory terms.
The frequencies of the oscillatory terms are given by the imaginary part of the
complex eigenvalues (Ruelle--Pollicott resonances). For large periods,
corrections to the well--known exponential growth of the smooth part of the
density are obtained. An alternative formula for rho(tau) in terms of the zeros
and poles of the Ruelle zeta function is also discussed. The results are
illustrated with the geodesic motion in billiards of constant negative
curvature. Connections with the statistical properties of the corresponding
quantum eigenvalues, random matrix theory and discrete maps are also
considered. In particular, a random matrix conjecture is proposed for the
eigenvalues of the classical evolution operator of chaotic billiards
Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds
In this paper, we numerically investigate the length spectra and the
low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large
number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero
eigenvalues have been successfully computed using the periodic orbit sum
method, which are compared with various geometric quantities such as volume,
diameter and length of the shortest periodic geodesic of the manifolds. The
deviation of low-lying eigenvalue spectra of manifolds converging to a cusped
hyperbolic manifold from the asymptotic distribution has been measured by
function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of
cusped manifolds in section 2 is correcte
Renormalization : A number theoretical model
We analyse the Dirichlet convolution ring of arithmetic number theoretic
functions. It turns out to fail to be a Hopf algebra on the diagonal, due to
the lack of complete multiplicativity of the product and coproduct. A related
Hopf algebra can be established, which however overcounts the diagonal. We
argue that the mechanism of renormalization in quantum field theory is modelled
after the same principle. Singularities hence arise as a (now continuously
indexed) overcounting on the diagonals. Renormalization is given by the map
from the auxiliary Hopf algebra to the weaker multiplicative structure, called
Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep
2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200
A Hierarchical Array of Integrable Models
Motivated by Harish-Chandra theory, we construct, starting from a simple
CDD\--pole \--matrix, a hierarchy of new \--matrices involving ever
``higher'' (in the sense of Barnes) gamma functions.These new \--matrices
correspond to scattering of excitations in ever more complex integrable
models.From each of these models, new ones are obtained either by
``\--deformation'', or by considering the Selberg-type Euler products of
which they represent the ``infinite place''. A hierarchic array of integrable
models is thus obtained. A remarkable diagonal link in this array is
established.Though many entries in this array correspond to familiar integrable
models, the array also leads to new models. In setting up this array we were
led to new results on the \--gamma function and on the \--deformed
Bloch\--Wigner function.Comment: 18 pages, EFI-92-2
Finite N Fluctuation Formulas for Random Matrices
For the Gaussian and Laguerre random matrix ensembles, the probability
density function (p.d.f.) for the linear statistic
is computed exactly and shown to satisfy a central limit theorem as . For the circular random matrix ensemble the p.d.f.'s for the linear
statistics and are calculated exactly by using a constant term identity
from the theory of the Selberg integral, and are also shown to satisfy a
central limit theorem as .Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho
Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws
We define a large class of continuous time multifractal random measures and
processes with arbitrary log-infinitely divisible exact or asymptotic scaling
law. These processes generalize within a unified framework both the recently
defined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and the
log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our
construction is based on some ``continuous stochastic multiplication'' from
coarse to fine scales that can be seen as a continuous interpolation of
discrete multiplicative cascades. We prove the stochastic convergence of the
defined processes and study their main statistical properties. The question of
genericity (universality) of limit multifractal processes is addressed within
this new framework. We finally provide some methods for numerical simulations
and discuss some specific examples.Comment: 24 pages, 4 figure
Symmetry Decomposition of Chaotic Dynamics
Discrete symmetries of dynamical flows give rise to relations between
periodic orbits, reduce the dynamics to a fundamental domain, and lead to
factorizations of zeta functions. These factorizations in turn reduce the labor
and improve the convergence of cycle expansions for classical and quantum
spectra associated with the flow. In this paper the general formalism is
developed, with the -disk pinball model used as a concrete example and a
series of physically interesting cases worked out in detail.Comment: CYCLER Paper 93mar01
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