194 research outputs found
Pizzetti formulae for Stiefel manifolds and applications
Pizzetti's formula explicitly shows the equivalence of the rotation invariant
integration over a sphere and the action of rotation invariant differential
operators. We generalize this idea to the integrals over real, complex, and
quaternion Stiefel manifolds in a unifying way. In particular we propose a new
way to calculate group integrals and try to uncover some algebraic structures
which manifest themselves for some well-known cases like the Harish-Chandra
integral. We apply a particular case of our formula to an Itzykson-Zuber
integral for the coset SO(4)/[SO(2)xSO(2)]. This integral naturally appears in
the calculation of the two-point correlation function in the transition of the
statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in
random matrix theory
Eigenvalue and Eigenvector Statistics in Time Series Analysis
The study of correlated time-series is ubiquitous in statistical analysis,
and the matrix decomposition of the cross-correlations between time series is a
universal tool to extract the principal patterns of behavior in a wide range of
complex systems. Despite this fact, no general result is known for the
statistics of eigenvectors of the cross-correlations of correlated time-series.
Here we use supersymmetric theory to provide novel analytical results that will
serve as a benchmark for the study of correlated signals for a vast community
of researchers.Comment: 8 pages, 3 figure
Chiral condensate and Dirac spectrum of one- and two-flavor QCD at nonzero -angle
In the -domain of QCD we have obtained exact analytical expressions
for the eigenvalue density of the Dirac operator at fixed for
both one and two flavors. These results made it possible to explain how the
different contributions to the spectral density conspire to give a chiral
condensate at fixed that does not change sign when the quark mass (or
one of the quark masses for two flavors) crosses the imaginary axis, while the
chiral condensate at fixed topological charge does change sign. From QCD at
nonzero density we have learnt that the discontinuity of the chiral condensate
may move to a different location when the spectral density increases
exponentially with the volume with oscillations on the order of the inverse
volume. This is indeed what happens when the product of the quark masses
becomes negative, but the situation is more subtle in this case: the
contribution of the "quenched" part of the spectral density diverges in the
thermodynamic limit at nonzero , but this divergence is canceled
exactly by the contribution from the zero modes. We conclude that the zero
modes are essential for the continuity of the chiral condensate and that their
contribution has to be perfectly balanced against the contribution from the
nonzero modes. Lattice simulations at nonzero -angle can only be
trusted if this is indeed the case.Comment: 9 pages, 8 figures, Contribution to the Proceedings of Lattice201
Universal distribution of Lyapunov exponents for products of Ginibre matrices
Starting from exact analytical results on singular values and complex
eigenvalues of products of independent Gaussian complex random
matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an
infinite product. We show that for a large number of product matrices the
distribution of each Lyapunov exponent is normal and compute its -dependent
variance as well as corrections in a expansion. Originally Lyapunov
exponents are defined for singular values of the product matrix that represents
a linear time evolution. Surprisingly a similar construction for the moduli of
the complex eigenvalues yields the very same exponents and normal distributions
to leading order. We discuss a general mechanism for matrices why
the singular values and the radii of complex eigenvalues collapse onto the same
value in the large- limit. Thereby we rederive Newman's triangular law which
has a simple interpretation as the radial density of complex eigenvalues in the
circular law and study the commutativity of the two limits and
on the global and the local scale. As a mathematical byproduct we
show that a particular asymptotic expansion of a Meijer G-function with large
index leads to a Gaussian.Comment: 36 pages, 6 figure
Chiral Random Matrix Theory: Generalizations and Applications
Kieburg M. Chiral Random Matrix Theory: Generalizations and Applications. Bielefeld: Fakultät für Physik; 2015
Products of Complex Rectangular and Hermitian Random Matrices
Products and sums of random matrices have seen a rapid development in the
past decade due to various analytical techniques available. Two of these are
the harmonic analysis approach and the concept of polynomial ensembles. Very
recently, it has been shown for products of real matrices with anti-symmetric
matrices of even dimension that the traditional harmonic analysis on matrix
groups developed by Harish-Chandra et al. needs to be modified when considering
the group action on general symmetric spaces of matrices. In the present work,
we consider the product of complex random matrices with Hermitian matrices, in
particular the former can be also rectangular while the latter has not to be
positive definite and is considered as a fixed matrix as well as a random
matrix. This generalises an approach for products involving the Gaussian
unitary ensemble (GUE) and circumvents the use there of non-compact group
integrals. We derive the joint probability density function of the real
eigenvalues and, additionally, prove transformation formulas for the
bi-orthogonal functions and kernels.Comment: 25 pages, v2: corrections of minor typos and an additional discussion
of Example IV.
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