14,849 research outputs found
Intriguing Patterns in the Roots of the Derivatives of some Random Polynomials
Our observations show that the sets of real (respectively complex) roots of the derivatives of some classical families of random polynomials admit a rich variety of patterns looking like discretized curves. To bring out the shapes of the suggested curves, we introduce an original use of fractional derivatives. Then we set several conjectures and outline a strategy to explain the presented phenomena. This strategy is based on asymptotic geometric properties of the corresponding complex critical points sets.Nos observations montrent que les enembles de racines reelles (resp. complexes) des derivees de certaines familles de polynomes aleatoires ont une riche varietes motifs structures qui ressemblent a des courbes discretisees. Pour faire clairement apparaitre ces courbes, nous avons recours a une utilisation originale des derivees fractionnaires. Nous posons ensuite des conjectures et proposons une strategie pour expliquer les phenomenes observes. Celle-ci est basee sur des proprietes de symetrie asymptotique de l'ensemble des points critiques de nos polynomes quand leur degre tend vers l'infini
Approximating the Permanent of a Random Matrix with Vanishing Mean
We show an algorithm for computing the permanent of a random matrix with
vanishing mean in quasi-polynomial time. Among special cases are the Gaussian,
and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we
can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time
2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the
intuition that the permanent is hard because of the "sign problem" - namely the
interference between entries of a matrix with different signs. A major open
question then remains whether one can provide an efficient algorithm for random
matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the
baseline assumptions of the BosonSampling paradigm
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]
The kissing polynomials and their Hankel determinants
We study a family of polynomials that are orthogonal with respect to the
weight function in , where . Since this
weight function is complex-valued and, for large , highly oscillatory,
many results in the classical theory of orthogonal polynomials do not apply. In
particular, the polynomials need not exist for all values of the parameter
, and, once they do, their roots lie in the complex plane. Our results
are based on analysing the Hankel determinants of these polynomials,
reformulated in terms of high-dimensional oscillatory integrals which are
amenable to asymptotic analysis. This analysis yields existence of the
even-degree polynomials for large values of , an asymptotic expansion
of the polynomials in terms of rescaled Laguerre polynomials near and a
description of the intricate structure of the roots of the Hankel determinants
in the complex plane. This work is motivated by the design of efficient
quadrature schemes for highly oscillatory integrals.Comment: 31 pages, 11 figures. Revised version, Section 8 edite
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