We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the conductivity and hence lead to anomalies in Drude's formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner's n-orbital model as well as the Anderson model in coherent potential approximation. 1 Introduction 1.1 Anomalous electronic transport Quantum effects and interactions in various materials cause a great variety of behaviors for electronic transport at low temperature. Understanding why some materials are conductors and others insulators is a challenging central problem of solid state physics. The first attempt to get a microscopic theory of electronic transport goes back to the...

We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of spectral measures, and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds involve the Hausdorff dimension of the spectral measure, multiplied by a correction calculated from the dynamical entropy, the density of states, and the capacity of the support. For Julia matrices, the correction disappears and the growth is ruled by the fractal dimension.

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.

We give an interpretation of the q-Catalan numbers in frameworks of the random matrix theory and weighted partitions of the set of integers. Key words: random matrices, Catalan numbers, non-commutative random variables AMS subject classification Primary: 15A52 Secondary: 05A18 In a joint discussion [3], Christian Mazza asked, what one can obtain when regarding the weighted pairings of 2k points under the condition that they are non-crossing? In the present note we give one possible answer to this question. Let us consider a set of N-dimensional random matrices A (r) , r ∈ N determined on the same probability space. We assume that these matrices are real symmetric and [A (r) N]ij = 1 √ a

We treat the stochastic equation for large N master fields proposed by Greensite and Halpern using a construction of master fields modelled after work of Voiculescu, and show that it contains the same information as the usual factorized Schwinger-Dyson equations. We comment on the relation to earlier work of Haan and of Cvitanovic, Lauwers and Scharbach. November