4,485 research outputs found
Pfaffian Expressions for Random Matrix Correlation Functions
It is well known that Pfaffian formulas for eigenvalue correlations are
useful in the analysis of real and quaternion random matrices. Moreover the
parametric correlations in the crossover to complex random matrices are
evaluated in the forms of Pfaffians. In this article, we review the
formulations and applications of Pfaffian formulas. For that purpose, we first
present the general Pfaffian expressions in terms of the corresponding skew
orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's
determinant formula for hermitian matrix models and explain how the evaluation
is simplified in the cases related to the classical orthogonal polynomials.
Applications of Pfaffian formulas to random matrix theory and other fields are
also mentioned.Comment: 28 page
Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
For the orthogonal-unitary and symplectic-unitary transitions in random
matrix theory, the general parameter dependent distribution between two sets of
eigenvalues with two different parameter values can be expressed as a
quaternion determinant. For the parameter dependent Gaussian and Laguerre
ensembles the matrix elements of the determinant are expressed in terms of
corresponding skew-orthogonal polynomials, and their limiting value for
infinite matrix dimension are computed in the vicinity of the soft and hard
edges respectively. A connection formula relating the distributions at the hard
and soft edge is obtained, and a universal asymptotic behaviour of the two
point correlation is identified.Comment: 37 pgs., 1fi
Recommended from our members
Complex networks
This chapter contains a brief introduction to complex networks, and in particular to small world and scale free networks. We show how to apply the replica method developed to analyse random matrices in statistical physics to calculate the spectral densities of the adjacency and Laplacian matrices of a scale free network. We use the effective medium approximation to treat networks with finite mean degree and discuss the local properties of random matrices associated with complex networks
Lattice Dirac fermions in a non-Abelian random gauge potential: Many flavors, chiral symmetry restoration and localization
In the previous paper we studied Dirac fermions in a non-Abelian random
vector potential by using lattice supersymmetry. By the lattice regularization,
the system of disordered Dirac fermions is defined without any ambiguities. We
showed there that at strong-disorder limit correlation function of the fermion
local density of states decays algebraically at the band center. In this paper,
we shall reexamine the multi-flavor or multi-species case rather in detail and
argue that the correlator at the band center decays {\em exponentially} for the
case of a {\em large} number of flavors. This means that a
delocalization-localization phase transition occurs as the number of flavors is
increased. This discussion is supported by the recent numerical studies on
multi-flavor QCD at the strong-coupling limit, which shows that the phase
structure of QCD drastically changes depending on the number of flavors. The
above behaviour of the correlator of the random Dirac fermions is closely
related with how the chiral symmetry is realized in QCD.Comment: Version appears in Mod.Phys.Lett.A17(2002)135
A theory of the electric quadrupole contribution to resonant x-ray scattering: Application to multipole ordering phases in Ce_{1-x}La_{x}B_{6}
We study the electric quadrupole (E2) contribution to resonant x-ray
scattering (RXS). Under the assumption that the rotational invariance is
preserved in the Hamiltonian describing the intermediate state of scattering,
we derive a useful expression for the RXS amplitude. One of the advantages the
derived expression possesses is the full information of the energy dependence,
lacking in all the previous studies using the fast collision approximation. The
expression is also helpful to classify the spectra into multipole order
parameters which are brought about. The expression is suitable to investigate
the RXS spectra in the localized f electron systems. We demonstrate the
usefulness of the formula by calculating the RXS spectra at the Ce L_{2,3}
edges in Ce_{1-x}La_{x}B_{6} on the basis of the formula. We obtain the spectra
as a function of energy in agreement with the experiment of
Ce_{0.7}La_{0.3}B_{6}. Analyzing the azimuthal angle dependence, we find the
sixfold symmetry in the \sigma-\sigma' channel and the threefold onein the
\sigma-\pi' channel not only in the antiferrooctupole (AFO) ordering phase but
also in the antiferroquadrupole (AFQ) ordering phase, which behavior depends
strongly on the domain distribution. The sixfold symmetry in the AFQ phase
arises from the simultaneously induced hexadecapole order. Although the AFO
order is plausible for phase IV in Ce_{1-x}La_{x}B_{6}, the possibility of the
AFQ order may not be ruled out on the basis of azimuthal angle dependence
alone.Comment: 12 pages, 6 figure
- âŠ