The Arithmetic of Distributions in Free Probability Theory

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\bold M$ contains either indecomposable ("prime") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free additive and multiplicative convolution semigroups the class $I_0$ consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in $\bold M$.Comment: 66 pages; latex; 5 figures; corrected version of proofs of Khintchine type theorems. For details see end of introductio

Colloidal glass transition: Beyond mode-coupling theory

A new theory for dynamics of concentrated colloidal suspensions and the colloidal glass transition is proposed. The starting point is the memory function representation of the density correlation function. The memory function can be expressed in terms of a time-dependent pair-density correlation function. An exact, formal equation of motion for this function is derived and a factorization approximation is applied to its evolution operator. In this way a closed set of equations for the density correlation function and the memory function is obtained. The theory predicts an ergodicity breaking transition similar to that predicted by the mode-coupling theory, but at a higher density.Comment: to be published in PR

Concentration of empirical distribution functions with applications to non-i.i.d. models

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Convergence to Stable Laws in Relative Entropy

Convergence to stable laws in relative entropy is established for sums of i.i.d. random variables

Mode Coupling relaxation scenario in a confined glass former

Molecular dynamics simulations of a Lennard-Jones binary mixture confined in a disordered array of soft spheres are presented. The single particle dynamical behavior of the glass former is examined upon supercooling. Predictions of mode coupling theory are satisfied by the confined liquid. Estimates of the crossover temperature are obtained by power law fit to the diffusion coefficients and relaxation times of the late $\alpha$ region. The $b$ exponent of the von Schweidler law is also evaluated. Similarly to the bulk, different values of the exponent $\gamma$ are extracted from the power law fit to the diffusion coefficients and relaxation times.Comment: 5 pages, 4 figures, changes in the text, accepted for publication on Europhysics Letter

Structural relaxation in orthoterphenyl: a schematic mode coupling theory model analysis

Depolarized light scattering spectra of orthoterphenyl showing the emergence of the structural relaxation below the oscillatory microscopic excitations are described by solutions of a schematic mode--coupling--theory model