672 research outputs found
Universal microscopic correlation functions for products of independent Ginibre matrices
We consider the product of n complex non-Hermitian, independent random
matrices, each of size NxN with independent identically distributed Gaussian
entries (Ginibre matrices). The joint probability distribution of the complex
eigenvalues of the product matrix is found to be given by a determinantal point
process as in the case of a single Ginibre matrix, but with a more complicated
weight given by a Meijer G-function depending on n. Using the method of
orthogonal polynomials we compute all eigenvalue density correlation functions
exactly for finite N and fixed n. They are given by the determinant of the
corresponding kernel which we construct explicitly. In the large-N limit at
fixed n we first determine the microscopic correlation functions in the bulk
and at the edge of the spectrum. After unfolding they are identical to that of
the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic
correlations we find at the origin differ for each n>1 and generalise the known
Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.Comment: 20 pages, v2 published version: typos corrected and references adde
Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
The location and asymptotic behaviour for large n of the zeros of exceptional
Jacobi and Laguerre polynomials are discussed. The zeros of exceptional
polynomials fall into two classes: the regular zeros, which lie in the interval
of orthogonality and the exceptional zeros, which lie outside that interval. We
show that the regular zeros have two interlacing properties: one is the natural
interlacing between consecutive polynomials as a consequence of their
Sturm-Liouville character, while the other one shows interlacing between the
zeros of exceptional and classical polynomials. A generalization of the
classical Heine-Mehler formula is provided for the exceptional polynomials,
which allows to derive the asymptotic behaviour of their regular zeros. We also
describe the location and the asymptotic behaviour of the exceptional zeros,
which converge for large n to fixed values.Comment: 19 pages, 3 figures, typed in AMS-LaTe
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
- …