Statistical mechanics of random graphs

We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.Comment: 6 pages (Talk at Conference: Applications of Physics in Financial Analysis 4, Warsaw 13-15 Nov. 2003

Spectral properties of empirical covariance matrices for data with power-law tails

We present an analytic method for calculating spectral densities of empirical covariance matrices for correlated data. In this approach the data is represented as a rectangular random matrix whose columns correspond to sampled states of the system. The method is applicable to a class of random matrices with radial measures including those with heavy (power-law) tails in the probability distribution. As an example we apply it to a multivariate Student distribution.Comment: 9 pages, 3 figures, references adde

Universality of random matrix dynamics

We discuss the concept of width-to-spacing ratio which plays the central role in the description of local spectral statistics of evolution operators in multiplicative and additive stochastic processes for random matrices. We show that the local spectral properties are highly universal and depend on a single parameter being the width-to-spacing ratio. We discuss duality between the kernel for Dysonian Brownian motion and the kernel for the Lyapunov matrix for the product of Ginibre matrices.Comment: 15 pages, 3 figure

Product Market Regulation and Labor Market Outcomes: How can Deregulation Create Jobs?

This paper reports on ongoing research on the interactions between product regulation and labor market outcomes. In particular, I summarize work on the employment effects of shop-closing regulation in the retail and other related sectors. Evidence on employment in the retail sector from Germany, the Netherlands and the United States suggests that the regulatory regime might play an important role; I argue that a nonnegligible comp o nent of the recent Dutch employment miracle could be attributed to product market deregulation, in particular liberalization of shop-closing laws effected in the mid-1990s. I sketch a model, based on Burda and Weil (1999), which can rationalize potential public interest aspects of such regulations as well as identify their employment and output costs.Product market regulation, retail trade, employment

Branching Data for Algebraic Functions and Representability by Radicals

The branching data of an algebraic function is a list of orders of local monodromies around branching points. We present branching data that ensure that the algebraic functions having them are representable by radicals. This paper is a review of recent work by the authors and of closely related classical work by Ritt.Comment: Submitted for publication to Banach Center Publications on April 1st, 201

Collapse of 4D random geometries

We extend the analysis of the Backgammon model to an ensemble with a fixed number of balls and a fluctuating number of boxes. In this ensemble the model exhibits a first order phase transition analogous to the one in higher dimensional simplicial gravity. The transition relies on a kinematic condensation and reflects a crisis of the integration measure which is probably a part of the more general problem with the measure for functional integration over higher (d>2) dimensional Riemannian structures.Comment: 7 pages, Latex2e, 2 figures (.eps

Quaternionic R transform and non-hermitian random matrices

Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B $\textbf{501}$, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of $X$ and its hermitian conjugate $X^\dagger$: \langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots \rangle\rangle for $N\rightarrow \infty$. We show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map $\mathcal{R}(z+wj) = x + \sigma^2 \left(\mu e^{2i\phi} z + w j\right)$ where $(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion $q=(z,w)\equiv z+ wj$. This map has five real parameters $\Re e x$, $\Im m x$, $\phi$, $\sigma$ and $\mu$. We use the R transform to calculate the limiting eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure