Asymptotic behavior of the distribution functions of local quantities in disordered conductors is studied in the weak disorder limit by means of an optimal fluctuation method. It is argued that this method is more appropriate for the study of seldom occurring events than the approaches based on nonlinear σ-models because it is capable of correctly handling fluctuations of the random potential with large amplitude as well as the short-scale structure of the corresponding solutions of the Schrödinger equation. For two-and three-dimensional conductors new asymptotics of the distribution functions are obtained which in some cases differ significantly from previously established results.

We study a class of network growth models with attachment rules governed by intrinsic node fitness. Both the individual node degree distribution and the degree correlation properties of the network are obtained as functions of the network growth rules. We also find analytical solutions to the inverse, design, problems of matching the growth rules to the required (e.g., power-law) node degree distribution and more generally to the required degree correlation function. We find that the design problems do not always have solutions. Among the specific conditions on the existence of solutions to the design problems is the requirement that the node degree distribution has to be broader than a certain threshold and the fact that factorizability of the correlation functions requires singular distributions of the node fitnesses. More generally, the restrictions on the input distributions and correlations that ensure solvability of the design problems are expressed in terms of the analytical properties of their generating functions.

We explore the influence of external perturbations on the energy levels of a Hamiltonian drawn at random from the Gaussian unitary distribution of Hermitian matrices. By deriving the joint distribution function of eigenvalues, we obtain the (n,m)-point parametric correlation function of the initial and the final density of states for perturbations of arbitrary rank and strength. A further generalization of these results allows for the incorporation of short-range spatial correlations in diffusive as well as ballistic chaotic structures.

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain. DOI: 10.1103/PhysRevLett.100.190404 PACS numbers: 03.65.Sq, 03.65.Yz, 05.30.Ch, 05.45.Mt Fidelity decay presently attracts considerable attention [1]. It measures the change of quantum dynamics of a state under a modification of the Hamiltonian. In quantum information, fidelity measures the deviation between a mathematical algorithm and its physical implementation. From a different point of view, important insight into the properties of the underlying systems is provided by the studies of correlations between spectra of random and/or chaotic Hamiltonians which differ by a parameterdependent perturbation [2]. Since statistical properties of fidelity decay in random or chaotic systems involve both spectra and eigenfunctions of the original and perturbed Hamiltonians, the existence of any connections between fidelity and purely spectral correlations is not a priori obvious.Random matrix theory (RMT) has been successful in describing quantum many-body systems and as a model for the spectral properties of single particle systems whose classical analogue is chaotic [3]. Within RMT fidelity was analyzed in linear response approximation [4] and both fidelity [5][6][7] and parametric correlations [8] were calculated exactly using the supersymmetry method. An unexpected fidelity revival at Heisenberg time was encountered [5] within RMT and confirmed in a dynamical coupled spin chain model [9].Earlier, differential relations between parametric spectral correlations and parametric density correlations were established [10,11]. By relating the latter to the fidelity amplitude via Fourier transform, we show in this Letter that the existence of these relations opens a crucial insight into the properties of fidelity decay. By analyzing the characteristic features of the parametric correlations in the time domain, the cross-form-factor, we discover a new, simple interpretation of the previously puzzling phenomenon of revival [5]. These relations follow directly from the basic definitions and symmetries of the underlying matrix models, being essentially Ward identities. We show that they are valid under very general assumptions. No explicit (e.g., supersymmetric) calculation is required; however, they rely on the universality of the parametric spectral correlations at the scale of mean level spacing. We thus explain the origin of various relations connecting spectral and wave-function correlations, and establish a unified framework for their analysis and generalizations. A relat...

We explore the influence of an arbitrary external potential perturbation V on the spectral properties of a weakly disordered conductor. In the framework of a statistical field theory of a nonlinear sigma-model type we find, depending on the range and the profile of the external perturbation, two qualitatively different universal regimes of parametric spectral statistics (i.e., cross correlations between the spectra of Hamiltonians H and H+V). We identify the translational invariance of the correlations in the space of Hamiltonians as the key indicator of universality, and find the connection between the coordinate system in this space which makes the translational invariance manifest, and the physically measurable properties of the system. In particular, in the case of localized perturbations, the latter turn out to be the eigenphases of the scattering matrix for scattering off the perturbing potential V. They also have a purely statistical interpretation in terms of the moments of the level velocity distribution. Finally, on the basis of this analysis, a set of results obtained recently by the authors using random matrix theory methods is shown to be applicable to a much wider class of disordered and chaotic structures.