17,238 research outputs found
Quantum Chaotic Dynamics and Random Polynomials
We investigate the distribution of roots of polynomials of high degree with
random coefficients which, among others, appear naturally in the context of
"quantum chaotic dynamics". It is shown that under quite general conditions
their roots tend to concentrate near the unit circle in the complex plane. In
order to further increase this tendency, we study in detail the particular case
of self-inversive random polynomials and show that for them a finite portion of
all roots lies exactly on the unit circle. Correlation functions of these roots
are also computed analytically, and compared to the correlations of eigenvalues
of random matrices. The problem of ergodicity of chaotic wave-functions is also
considered. For that purpose we introduce a family of random polynomials whose
roots spread uniformly over phase space. While these results are consistent
with random matrix theory predictions, they provide a new and different insight
into the problem of quantum ergodicity. Special attention is devoted all over
the paper to the role of symmetries in the distribution of roots of random
polynomials.Comment: 33 pages, Latex, 6 Figures not included (a copy of them can be
requested at [email protected]); to appear in Journal of Statistical
Physic
Distribution of roots of random real generalized polynomials
The average density of zeros for monic generalized polynomials,
, with real holomorphic and
real Gaussian coefficients is expressed in terms of correlation functions of
the values of the polynomial and its derivative. We obtain compact expressions
for both the regular component (generated by the complex roots) and the
singular one (real roots) of the average density of roots. The density of the
regular component goes to zero in the vicinity of the real axis like
. We present the low and high disorder asymptotic
behaviors. Then we particularize to the large limit of the average density
of complex roots of monic algebraic polynomials of the form with real independent, identically distributed
Gaussian coefficients having zero mean and dispersion . The average density tends to a simple, {\em universal}
function of and in the domain where nearly all the roots are located for
large .Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed
tarfile (.66MB) containing 8 Postscript figures is available by e-mail from
[email protected]
Real roots of Random Polynomials: Universality close to accumulation points
We identify the scaling region of a width O(n^{-1}) in the vicinity of the
accumulation points of the real roots of a random Kac-like polynomial
of large degree n. We argue that the density of the real roots in this region
tends to a universal form shared by all polynomials with independent,
identically distributed coefficients c_i, as long as the second moment
\sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to
the previously reported abrupt) and quite nontrivial suppression of the number
of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled
as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been
removed. 10 pages, 12 eps figures. This version contains all updates, clearer
pictures and some more thorough explanation
Random polynomials, random matrices, and -functions
We show that the Circular Orthogonal Ensemble of random matrices arises
naturally from a family of random polynomials. This sheds light on the
appearance of random matrix statistics in the zeros of the Riemann
zeta-function.Comment: Added background material. Final version. To appear in Nonlinearit
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