16,882 research outputs found
The supersymmetric technique for random-matrix ensembles with zero eigenvalues
The supersymmetric technique is applied to computing the average spectral
density near zero energy in the large-N limit of the random-matrix ensembles
with zero eigenvalues: B, DIII-odd, and the chiral ensembles (classes AIII,
BDI, and CII). The supersymmetric calculations reproduce the existing results
obtained by other methods. The effect of zero eigenvalues may be interpreted as
reducing the symmetry of the zero-energy supersymmetric action by breaking a
certain abelian symmetry.Comment: 22 pages, introduction modified, one reference adde
A class of nonsymmetric preconditioners for saddle point problems
For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric
Universality of correlation functions of hermitian random matrices in an external field
The behavior of correlation functions is studied in a class of matrix models
characterized by a measure containing a potential term and an
external source term: S=N\tr(V(M)-MA). In the large limit, the
short-distance behavior is found to be identical to the one obtained in
previously studied matrix models, thus extending the universality of the
level-spacing distribution. The calculation of correlation functions involves
(finite ) determinant formulae, reducing the problem to the large
asymptotic analysis of a single kernel . This is performed by an appropriate
matrix integral formulation of . Multi-matrix generalizations of these
results are discussed.Comment: 29 pages, Te
Characteristic polynomials of random matrices
Number theorists have studied extensively the connections between the
distribution of zeros of the Riemann -function, and of some
generalizations, with the statistics of the eigenvalues of large random
matrices. It is interesting to compare the average moments of these functions
in an interval to their counterpart in random matrices, which are the
expectation values of the characteristic polynomials of the matrix. It turns
out that these expectation values are quite interesting. For instance, the
moments of order 2K scale, for unitary invariant ensembles, as the density of
eigenvalues raised to the power ; the prefactor turns out to be a
universal number, i.e. it is independent of the specific probability
distribution. An equivalent behaviour and prefactor had been found, as a
conjecture, within number theory. The moments of the characteristic
determinants of random matrices are computed here as limits, at coinciding
points, of multi-point correlators of determinants. These correlators are in
fact universal in Dyson's scaling limit in which the difference between the
points goes to zero, the size of the matrix goes to infinity, and their product
remains finite.Comment: 30 pages,late
Stability Estimates and Structural Spectral Properties of Saddle Point Problems
For a general class of saddle point problems sharp estimates for
Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants
in Brezzi's theory. In the finite-dimensional Hermitian case more detailed
spectral properties of preconditioned saddle point matrices are presented,
which are helpful for the convergence analysis of common Krylov subspace
methods. The theoretical results are applied to two model problems from optimal
control with time-periodic state equations. Numerical experiments with the
preconditioned minimal residual method are reported
- …