503 research outputs found
Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry
The class of norm-dependent Random Matrix Ensembles is studied in the
presence of an external field. The probability density in those ensembles
depends on the trace of the squared random matrices, but is otherwise
arbitrary. An exact mapping to superspace is performed. A transformation
formula is derived which gives the probability density in superspace as a
single integral over the probability density in ordinary space. This is done
for orthogonal, unitary and symplectic symmetry. In the case of unitary
symmetry, some explicit results for the correlation functions are derived.Comment: 19 page
Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry
We generalize the supersymmetry method in Random Matrix Theory to arbitrary
rotation invariant ensembles. Our exact approach further extends a previous
contribution in which we constructed a supersymmetric representation for the
class of norm-dependent Random Matrix Ensembles. Here, we derive a
supersymmetric formulation under very general circumstances. A projector is
identified that provides the mapping of the probability density from ordinary
to superspace. Furthermore, it is demonstrated that setting up the theory in
Fourier superspace has considerable advantages. General and exact expressions
for the correlation functions are given. We also show how the use of hyperbolic
symmetry can be circumvented in the present context in which the non-linear
sigma model is not used. We construct exact supersymmetric integral
representations of the correlation functions for arbitrary positions of the
imaginary increments in the Green functions.Comment: 36 page
Transition from Poisson to gaussian unitary statistics: The two-point correlation function
We consider the Rosenzweig-Porter model of random matrix which interpolates
between Poisson and gaussian unitary statistics and compute exactly the
two-point correlation function. Asymptotic formulas for this function are given
near the Poisson and gaussian limit.Comment: 19 pages, no figure
Quantile Correlations: Uncovering temporal dependencies in financial time series
We conduct an empirical study using the quantile-based correlation function
to uncover the temporal dependencies in financial time series. The study uses
intraday data for the S\&P 500 stocks from the New York Stock Exchange. After
establishing an empirical overview we compare the quantile-based correlation
function to stochastic processes from the GARCH family and find striking
differences. This motivates us to propose the quantile-based correlation
function as a powerful tool to assess the agreements between stochastic
processes and empirical data
Spin-orbit interaction in quantum dots in the presence of exchange correlations
We discuss the problem of spin-orbit interaction in a 2D chaotic or diffusive
quantum dot in the presence of exchange correlations. Spin-orbit scattering
breaks spin rotation invariance, and in the crossover regime between different
symmetries of the spin-orbit coupling, the problem has no closed solution. A
conventional choice of a many-particle basis in a numerical diagonalization is
the set of Slater determinants built from the single-particle eigenstates of
the one-body Hamiltonian (including the spin-orbit terms). We develop a
different approach based on the use of a good-spin many-particle basis that is
composed of the eigenstates of the universal Hamiltonian in the absence of
spin-orbit scattering. We introduce a complete labelling of this good-spin
basis and use angular momentum algebra to calculate in closed form the matrix
elements of the spin-orbit interaction in this basis. Spin properties, such as
the ground-state spin distribution and the spin excitation function, are easily
calculated in this basis.Comment: 14 pages, 3 figure
Rigidity and Normal Modes in Random Matrix Spectra
We consider the Gaussian ensembles of random matrices and describe the normal
modes of the eigenvalue spectrum, i.e., the correlated fluctuations of
eigenvalues about their most probable values. The associated normal mode
spectrum is linear, and for large matrices, the normal modes are found to be
Chebyshev polynomials of the second kind. We contrast this with the behaviour
of a sequence of uncorrelated levels, which has a quadratic normal mode
spectrum. The difference in the rigidity of random matrix spectra and sequences
of uncorrelated levels can be attributed to this difference in the normal mode
spectra. We illustrate this by calculating the number variance in the two
cases.Comment: 12 pages, 1 LaTeX fil
Stochastic field theory for a Dirac particle propagating in gauge field disorder
Recent theoretical and numerical developments show analogies between quantum
chromodynamics (QCD) and disordered systems in condensed matter physics. We
study the spectral fluctuations of a Dirac particle propagating in a finite
four dimensional box in the presence of gauge fields. We construct a model
which combines Efetov's approach to disordered systems with the principles of
chiral symmetry and QCD. To this end, the gauge fields are replaced with a
stochastic white noise potential, the gauge field disorder. Effective
supersymmetric non-linear sigma-models are obtained. Spontaneous breaking of
supersymmetry is found. We rigorously derive the equivalent of the Thouless
energy in QCD. Connections to other low-energy effective theories, in
particular the Nambu-Jona-Lasinio model and chiral perturbation theory, are
found.Comment: 4 pages, 1 figur
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