634 research outputs found
Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
It is known that the moments of the maximum value of a one-dimensional
conditional Brownian motion, the three-dimensional Bessel bridge with duration
1 started from the origin, are expressed using the Riemann zeta function. We
consider a system of two Bessel bridges, in which noncolliding condition is
imposed. We show that the moments of the maximum value is then expressed using
the double Dirichlet series, or using the integrals of products of the Jacobi
theta functions and its derivatives. Since the present system will be provided
as a diffusion scaling limit of a version of vicious walker model, the ensemble
of 2-watermelons with a wall, the dominant terms in long-time asymptotics of
moments of height of 2-watermelons are completely determined. For the height of
2-watermelons with a wall, the average value was recently studied by Fulmek by
a method of enumerative combinatorics.Comment: v2: LaTeX, 19 pages, 2 figures, minor corrections made for
publication in J. Stat. Phy
Beyond universality in random matrix theory
In order to have a better understanding of finite random matrices with
non-Gaussian entries, we study the expansion of local eigenvalue
statistics in both the bulk and at the hard edge of the spectrum of random
matrices. This gives valuable information about the smallest singular value not
seen in universality laws. In particular, we show the dependence on the fourth
moment (or the kurtosis) of the entries. This work makes use of the so-called
complex Gaussian divisible ensembles for both Wigner and sample covariance
matrices.Comment: Published at http://dx.doi.org/10.1214/15-AAP1129 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
Numerical calculation of Bessel, Hankel and Airy functions
The numerical evaluation of an individual Bessel or Hankel function of large
order and large argument is a notoriously problematic issue in physics.
Recurrence relations are inefficient when an individual function of high order
and argument is to be evaluated. The coefficients in the well-known uniform
asymptotic expansions have a complex mathematical structure which involves Airy
functions. For Bessel and Hankel functions, we present an adapted algorithm
which relies on a combination of three methods: (i) numerical evaluation of
Debye polynomials, (ii) calculation of Airy functions with special emphasis on
their Stokes lines, and (iii) resummation of the entire uniform asymptotic
expansion of the Bessel and Hankel functions by nonlinear sequence
transformations.
In general, for an evaluation of a special function, we advocate the use of
nonlinear sequence transformations in order to bridge the gap between the
asymptotic expansion for large argument and the Taylor expansion for small
argument ("principle of asymptotic overlap"). This general principle needs to
be strongly adapted to the current case, taking into account the complex phase
of the argument. Combining the indicated techniques, we observe that it
possible to extend the range of applicability of existing algorithms. Numerical
examples and reference values are given.Comment: 18 pages; 7 figures; RevTe
- …