4,389 research outputs found
Periodic orbits of the ensemble of Sinai-Arnold cat maps and pseudorandom number generation
We propose methods for constructing high-quality pseudorandom number
generators (RNGs) based on an ensemble of hyperbolic automorphisms of the unit
two-dimensional torus (Sinai-Arnold map or cat map) while keeping a part of the
information hidden. The single cat map provides the random properties expected
from a good RNG and is hence an appropriate building block for an RNG, although
unnecessary correlations are always present in practice. We show that
introducing hidden variables and introducing rotation in the RNG output,
accompanied with the proper initialization, dramatically suppress these
correlations. We analyze the mechanisms of the single-cat-map correlations
analytically and show how to diminish them. We generalize the Percival-Vivaldi
theory in the case of the ensemble of maps, find the period of the proposed RNG
analytically, and also analyze its properties. We present efficient practical
realizations for the RNGs and check our predictions numerically. We also test
our RNGs using the known stringent batteries of statistical tests and find that
the statistical properties of our best generators are not worse than those of
other best modern generators.Comment: 18 pages, 3 figures, 9 table
Statistics of two-dimensional random walks, the "cyclic sieving phenomenon" and the Hofstadter model
We focus on the algebraic area probability distribution of planar random
walks on a square lattice with , , and steps right, left,
up and down. We aim, in particular, at the algebraic area generating function
evaluated at Q=e^{2\i\pi\over q}, a root of unity,
when both and are multiples of . In the simple case of
staircase walks, a geometrical interpretation of
in terms of the cyclic sieving phenomenon is
illustrated. Then, an expression for , which is
relevant to the Stembridge's case, is proposed. Finally, the related problem of
evaluating the n-th moments of the Hofstadter Hamiltonian in the commensurate
case is addressed.Comment: 13 pages, LaTeX 2
A family of centered random walks on weight lattices conditioned to stay in Weyl chambers
Under a natural asumption on the drift, the law of the simple random walk on
the multidimensional first quadrant conditioned to always stay in the first
octant was obtained by O'Connell in [O]. It coincides with that of the image of
the simple random walk under the multidimensional Pitman transform and can be
expressed in terms of specializations of Schur functions. This result has been
generalized in [LLP1] and [LLP2] for a large class of random walks on weight
lattices defined from representations of Kac-Moody algebras and their
conditionings to always stay in Weyl chambers. In these various works, the
drift of the considered random walk is always assumed in the interior of the
cone. In this paper, we introduce for some zero drift random walks defined from
minuscule representations a relevant notion of conditioning to stay in Weyl
chambers and we compute their laws. Namely, we consider the conditioning for
these walks to stay in these cones until an instant we let tend to infinity. We
also prove that the laws so obtained can be recovered by letting the drift tend
to zero in the transitions matrices obtained in [LLP1]. We also conjecture our
results remain true in the more general case of a drift in the frontier of the
Weyl chamber
Exact Meander Asymptotics: a Numerical Check
This note addresses the meander enumeration problem: "Count all topologically
inequivalent configurations of a closed planar non self-intersecting curve
crossing a line through a given number of points". We review a description of
meanders introduced recently in terms of the coupling to gravity of a
two-flavored fully-packed loop model. The subsequent analytic predictions for
various meandric configuration exponents are checked against exact enumeration,
using a transfer matrix method, with an excellent agreement.Comment: 48 pages, 24 figures, tex, harvmac, eps
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