38 research outputs found
Permanental processes from products of complex and quaternionic induced Ginibre ensembles
We consider products of independent random matrices taken from the induced
Ginibre ensemble with complex or quaternion elements. The joint densities for
the complex eigenvalues of the product matrix can be written down exactly for a
product of any fixed number of matrices and any finite matrix size. We show
that the squared absolute values of the eigenvalues form a permanental process,
generalising the results of Kostlan and Rider for single matrices to products
of complex and quaternionic matrices. Based on these findings, we can first
write down exact results and asymptotic expansions for the so-called hole
probabilities, that a disc centered at the origin is void of eigenvalues.
Second, we compute the asymptotic expansion for the opposite problem, that a
large fraction of complex eigenvalues occupies a disc of fixed radius centered
at the origin; this is known as the overcrowding problem. While the expressions
for finite matrix size depend on the parameters of the induced ensembles, the
asymptotic results agree to leading order with previous results for products of
square Ginibre matrices.Comment: 47 pages, v2: typos corrected, 1 reference added, published versio
Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
\amp{\psi_f}{\psi}, where is a fixed state in Hilbert space
and is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space , where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established.Comment: 11 pages, 1 figure, revtex
Average characteristic polynomials in the two-matrix model
The two-matrix model is defined on pairs of Hermitian matrices of
size by the probability measure where
and are given potential functions and \tau\in\er. We study averages
of products and ratios of characteristic polynomials in the two-matrix model,
where both matrices and may appear in a combined way in both
numerator and denominator. We obtain determinantal expressions for such
averages. The determinants are constructed from several building blocks: the
biorthogonal polynomials and associated to the two-matrix
model; certain transformed functions and \Q_n(v); and finally
Cauchy-type transforms of the four Eynard-Mehta kernels , ,
and . In this way we generalize known results for the
-matrix model. Our results also imply a new proof of the Eynard-Mehta
theorem for correlation functions in the two-matrix model, and they lead to a
generating function for averages of products of traces.Comment: 28 pages, references adde
The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models
The k-point correlation functions of the Gaussian Random Matrix Ensembles are
certain determinants of functions which depend on only two arguments. They are
referred to as kernels, since they are the building blocks of all correlations.
We show that the kernels are obtained, for arbitrary level number, directly
from supermatrix models for one-point functions. More precisely, the generating
functions of the one-point functions are equivalent to the kernels. This is
surprising, because it implies that already the one-point generating function
holds essential information about the k-point correlations. This also
establishes a link to the averaged ratios of spectral determinants, i.e. of
characteristic polynomials
On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble
We consider the asymptotics of the correlation functions of the
characteristic polynomials of the hermitian Wigner matrices .
We show that for the correlation function of any even order the asymptotic
coincides with this for the GUE up to a factor, depending only on the forth
moment of the common probability law of entries , ,
i.e. that the higher moments of do not contribute to the above limit.Comment: 20
Berry's phase for compact Lie groups
The Lie group adiabatic evolution determined by a Lie algebra parameter
dependent Hamiltonian is considered. It is demonstrated that in the case when
the parameter space of the Hamiltonian is a homogeneous K\"ahler manifold its
fundamental K\"ahler potentials completely determine Berry geometrical phase
factor. Explicit expressions for Berry vector potentials (Berry connections)
and Berry curvatures are obtained using the complex parametrization of the
Hamiltonian parameter space. A general approach is exemplified by the Lie
algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.Comment: 24 pages, no figure
An exact formula for general spectral correlation function of random Hermitian matrices
We have found an exact formula expressing a general correlation function
containing both products and ratios of characteristic polynomials of random
Hermitian matrices. The answer is given in the form of a determinant. An
essential difference from the previously studied correlation functions (of
products only) is the appearance of non-polynomial functions along with the
orthogonal polynomials. These non-polynomial functions are the Cauchy
transforms of the orthogonal polynomials. The result is valid for any ensemble
of beta=2 symmetry class and generalizes recent asymptotic formulae obtained
for GUE and its chiral counterpart by different methods..Comment: published version, with a few misprints correcte
Real roots of Random Polynomials: Universality close to accumulation points
We identify the scaling region of a width O(n^{-1}) in the vicinity of the
accumulation points of the real roots of a random Kac-like polynomial
of large degree n. We argue that the density of the real roots in this region
tends to a universal form shared by all polynomials with independent,
identically distributed coefficients c_i, as long as the second moment
\sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to
the previously reported abrupt) and quite nontrivial suppression of the number
of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled
as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been
removed. 10 pages, 12 eps figures. This version contains all updates, clearer
pictures and some more thorough explanation