By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.
PACS 05.45.Df -Fractals PACS 71.30.+h -Metal-insulator transitions and other electronic transitions Abstract -Based on heuristic arguments we conjecture that an intimate relation exists between the eigenfunction multifractal dimensions Dq of the eigenstates of critical random matrix ensemblesWe verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility χ describing level correlations can be related to Dq in a unified way as Dq−1 , thus generalizing existing relations with relevance to the disorder driven Anderson-transition.Introduction. -It is well-known that the spatial fluctuations of the eigenstates in a disordered system at the Anderson-transition show multifractal characteristics [1,2] which has been demonstrated recently in a series of experiments [3]. Therefore the modeling and analysis of multifractal states has become of central importance producing many interesting results. For this purpose random matrix models have been invoked and studied recently [4][5][6].Since the exact, analytical prediction of the multifractal dimensions of the states for the experimentally relevant Anderson-transition in d = 3 or the integer quantum-Hall transition in d = 2 seems to be out of reach, it is desirable to search for heuristic relations in order to understand the complexity of the states at criticality. In the present paper we propose such heuristic relations that are numerically verified using various ensembles of random matrices.The spatial fluctuations of the eigenstates can be described by a set of multifractal dimensions D q defined by the scaling of the inverse mean eigenfunction participation numbers with the system size N :
Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, Dq and D q ′ , of the eigenstates of critical random matrix ensembles:Here, we verify this relation by extensive numerical calculations on critical random matrix ensembles and extend its applicability to q < 1/2 but also to deterministic models producing multifractal eigenstates and to generic multifractal structures. We also demonstrate, for the scattering version of the power-law banded random matrix model at criticality, that the scaling exponents σq of the inverse moments of Wigner delay times, τ
We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes N and the average connectivity α. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements <|S(mn)|(2)>, the conductance probability distribution w(T), the average conductance
We perform a detailed numerical study of the distribution of conductances P (T ) for quasi-onedimensional corrugated waveguides as a function of the corrugation complexity (from rough to smooth). We verify the universality of P (T ) in both, the diffusive ( T > 1) and the localized ( T ≪ 1) transport regimes. However, at the crossover regime ( T ∼ 1), we observe that P (T ) evolves from the surface-disorder to the bulk-disorder theoretical predictions for decreasing complexity in the waveguide boundaries. We explain this behavior as a transition from disorder to deterministic chaos; since, in the limit of smooth boundaries the corrugated waveguides are, effectively, linear chains of chaotic cavities.
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