110 research outputs found
Spectral Density of Complex Networks with a Finite Mean Degree
In order to clarify the statistical features of complex networks, the
spectral density of adjacency matrices has often been investigated. Adopting a
static model introduced by Goh, Kahng and Kim, we analyse the spectral density
of complex scale free networks. For that purpose, we utilize the replica method
and effective medium approximation (EMA) in statistical mechanics. As a result,
we identify a new integral equation which determines the asymptotic spectral
density of scale free networks with a finite mean degree . In the limit , known asymptotic formulae are rederived. Moreover, the
corrections to known results are analytically calculated by a perturbative
method.Comment: 18 pages, 1 figure, minor corrections mad
Correlation functions for random involutions
Our interest is in the scaled joint distribution associated with
-increasing subsequences for random involutions with a prescribed number of
fixed points. We proceed by specifying in terms of correlation functions the
same distribution for a Poissonized model in which both the number of symbols
in the involution, and the number of fixed points, are random variables. From
this, a de-Poissonization argument yields the scaled correlations and
distribution function for the random involutions. These are found to coincide
with the same quantities known in random matrix theory from the study of
ensembles interpolating between the orthogonal and symplectic universality
classes at the soft edge, the interpolation being due to a rank 1 perturbation.Comment: 27 pages, 1 figure, minor corrections mad
Vicious Random Walkers and a Discretization of Gaussian Random Matrix Ensembles
The vicious random walker problem on a one dimensional lattice is considered.
Many walkers take simultaneous steps on the lattice and the configurations in
which two of them arrive at the same site are prohibited. It is known that the
probability distribution of N walkers after M steps can be written in a
determinant form. Using an integration technique borrowed from the theory of
random matrices, we show that arbitrary k-th order correlation functions of the
walkers can be expressed as quaternion determinants whose elements are
compactly expressed in terms of symmetric Hahn polynomials.Comment: LaTeX, 15 pages, 1 figure, minor corrections made before publication
in Nucl. Phys.
Dynamical Correlations among Vicious Random Walkers
Nonintersecting motion of Brownian particles in one dimension is studied. The
system is constructed as the diffusion scaling limit of Fisher's vicious random
walk. N particles start from the origin at time t=0 and then undergo mutually
avoiding Brownian motion until a finite time t=T. In the short time limit , the particle distribution is asymptotically described by Gaussian
Unitary Ensemble (GUE) of random matrices. At the end time t = T, it is
identical to that of Gaussian Orthogonal Ensemble (GOE). The Brownian motion is
generally described by the dynamical correlations among particles at many times
between t=0 and t=T. We show that the most general dynamical
correlations among arbitrary number of particles at arbitrary number of times
are written in the forms of quaternion determinants. Asymptotic forms of the
correlations in the limit are evaluated and a discontinuous
transition of the universality class from GUE to GOE is observed.Comment: REVTeX3.1, 4 pages, no figur
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
Eigenvalue statistics of the real Ginibre ensemble
The real Ginibre ensemble consists of random matrices formed
from i.i.d. standard Gaussian entries. By using the method of skew orthogonal
polynomials, the general -point correlations for the real eigenvalues, and
for the complex eigenvalues, are given as Pfaffians with explicit
entries. A computationally tractable formula for the cumulative probability
density of the largest real eigenvalue is presented. This is relevant to May's
stability analysis of biological webs.Comment: 4 pages, to appear PR
Form Factor of a Quantum Graph in a Weak Magnetic Field
Using periodic orbit theory, we evaluate the form factor of a quantum graph
to which a very weak magnetic field is applied. The first correction to the
diagonal approximation describing the transition between the universality
classes is shown to be in agreement with Pandey and Mehta's formula of
parametric random matrix theory.Comment: LaTeX, 7 pages, no figur
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