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 $p$. In the limit $p \to \infty$, known asymptotic formulae are rederived. Moreover, the $1/p$ 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 $k$-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 $t \ll T$, 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 $t_1,t_2,..., t_M$ 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 $N \to \infty$ 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 $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the complex eigenvalues, are given as $n \times n$ 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