We study statistical properties of the ensemble of large NϫN random matrices whose entries H i j decrease in a power-law fashion H i j ϳ͉iϪ j͉ Ϫ␣ . Mapping the problem onto a nonlinear model with nonlocal interaction, we find a transition from localized to extended states at ␣ϭ1. At this critical value of ␣ the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at ␣ϭ1, parametrized by the value of the coupling constant of the model. At ␣Ͼ1 all states are expected to be localized with integrable power-law tails. At the same time, for 1Ͻ␣Ͻ3/2 the wave packet spreading at a short time scale is superdiffusive: ͉͗r͉͘ϳt 1/(2␣Ϫ1) , which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2Ͻ␣Ͻ1 the statistical properties of eigenstates are similar to those in a metallic sample in dϭ(␣Ϫ1/2) Ϫ1 dimensions. Finally, the region ␣Ͻ1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (␣ϭ0). The theoretical predictions are compared with results of numerical simulations.

We review recent progress in analysing wave scattering in systems with both intrinsic chaos and/or disorder and internal losses, when the scattering matrix is no longer unitary. By mapping the problem onto a nonlinear supersymmetric σmodel, we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system. One of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel. The developed theory has far-reaching consequences for many quantities of interest, including local Green functions and time delays. In particular, we point out the role played by absorption as a sensitive indicator of mechanisms behind the Anderson localisation transition. We also provide a random-matrix based analysis of S-matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken timereversal invariance, completing previous works on the subject. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas.

We investigate some implications of the freezing scenario proposed by Carpentier and Le Doussal (CLD) for a Random Energy Model (REM) with logarithmically correlated random potential. We introduce a particular (circular) variant of the model, and show that the integer moments of the partition function in the hightemperature phase are given by the well-known Dyson Coulomb gas integrals. The CLD freezing scenario allows one to use those moments for extracting the distribution of the free energy in both high-and low-temperature phases. In particular, it yields the full distribution of the minimal value in the potential sequence. This provides an explicit new class of extreme-value statistics for strongly correlated variables, manifestly different from the standard Gumbel class.

Abstract. We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. Our method extends the recent analysis of [13] from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a duality relation satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the [0, 1] interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.arXiv:0907.2359v2 [cond-mat.dis-nn]

We reduce a random-band-matrix (RBM) problem to a one-dimensional, nonlinear, supersymmetric, a model. This reduction becomes exact in the limit b-+°°,b being the effective bandwidth. We prove that b 2 /N, N being the matrix size, is the relevant scaling parameter. When the mean value of diagonal elements increases linearly along the diagonal an extra scaling parameter arises. These conclusions are in agreement with recent numerical results. PACS numbers: 05.45.-hb, 72.15.Rn Among other ensembles of random matrices that of band matrices is under intensive investigation at present. Random band matrices (RBM) were claimed to be relevant for the explanation of properties of quantum systems whose classical counterparts display chaotic behavior [1]. The kicked rotator should be mentioned as a generic example [2]. Besides, RBM arise in the course of investigation of the conductance fluctuations of thin disordered slabs by a transfer-matrix method [3], Physical applications of RBM dictate the concentration of interest on localization properties of their eigenvectors. When the bandwidth b is sufficiently small all eigenvectors are localized and eigenvalues turn out to be noncorrelated. This is the intrinsic property of spectra of quantum systems integrable in the classical limit [4]. In the opposite case bozN, it is quite clear that the RBM ensemble does not differ practically from the Gaussian one characterized by delocalized eigenvectors and correlated eigenvalues (modeling spectral properties of "chaotic" quantum systems [4]). Therefore, the RBM ensemble in a whole range of bandwidths \ 0 [9],In the present paper we perform the analytical investigation of the RBM properties within the framework of the supersymmetric approach. This method proved to be quite powerful as applied to Gaussian ensembles [10,11] and that of sparse random matrices [12]. By using the procedure analogous to that introduced by Schafer and Wegner [13], we reduce the RBM problem in the limit 7V^>1, b^>\ to the on...

We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via M open channels; a = 1, 2, .., M . A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the χ 2 distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer-Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases θa of the S-matrix (taken modulo 2π). We use it to find the distribution of derivatives τa = ∂θa/∂E of these eigenphases with respect to the energy ("partial delay times" ) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner-Smith time delay τw(E) = 1 M a ∂θa/∂E at two different energies E − Ω/2 and E + Ω/2 as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.

The article reviews recent analytical results concerning statistical properties of eigenfunctions of random Hamiltonians with broken time reversal symmetry describing a motion of a quantum particle in a thick wire of finite length L. It is demonstrated that the problem is equivalent to the study of properties of large Random Banded Matrices in the limit of large width of the band. Matrices of this class are relevant for a number of problems in Solid State physics and in the domain of Quantum Chaos. We find the analytical expressions for the distribution of the following quantities: i) the eigenfunction amplitude |ψ(r)|2 at given point of the sample; ii) spatial extent of the eigenfunction measured by the “inverse participation ratio” P=∫V dr|ψ(r)|4; iii) the quantity R=|ψ(r)ψ(r′)|2, points r and r′ belonging to the opposite ends of the sample. For a long sample the quantity –(ln R)/L characterizes the decay rate of a localized eigenfunction (Lyapunov exponent). Relation with available numerical results is discussed.