Disorder perturbed Flat Bands I: Level density and Inverse Participation Ratio

We consider the effect of disorder on the tight-binding Hamiltonians with a flat band and derive a common mathematical formulation of the average density of states and inverse participation ratio applicable for a wide range of them. The system information in the formulation appears through a single parameter which plays an important role in search of the critical points for disorder driven transitions in flat bands [1]. In weak disorder regime, the formulation indicates an insensitivity of the statistical measures to disorder strength, thus confirming the numerical results obtained by our as well as previous studies.Comment: 34 Pages, 4 figures, Previous version divided in two parts to include new sections and details, title and abstract changed to

Eigenfunction Statistics of Complex Systems: A Common Mathematical Formulation

We derive a common mathematical formulation for the eigenfunction statistics of Hermitian operators, represented by a multi-parametric probability density. The system-information in the formulation enters through two parameters only, namely, system size and the complexity parameter, a function of all system parameter including size. The behavior is contrary to the eigenvalue statistics which is sensitive to complexity parameter only and shows a single parametric scaling. The existence of a mathematical formulation, of both eigenfunctions and eigenvalues, common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties. This also suggests the possibility to classify them in various universality classes defined by complexity parameter.Comment: 16 Figures, Several Changes, Many new sections and figures included, conclusion slightly change

Level Correlations for Metal-Insulator Transition

We study the level-statistics of a disordered system undergoing the Anderson type metal-insulator transition. The disordered Hamiltonian is a sparse random matrix in the site representation and the statistics is obtained by taking an ensemble of such matrices. It is shown that the transition of levels due to change of various parameters e.g. disorder, system size, hopping rate can be mapped to the perturbation driven evolution of the eigenvalues of an ensemble subjected to Wigner-Dyson type perturbation with an initial state given by a Poisson ensemble; the mapping is then used to obtain desired level-correlations.Comment: Latex file, 7 pages, No figure

On the Distribution of zeros of chaotic wavefunction

The wavefunctions in phase-space representation can be expressed as entire functions of their zeros if the phase space is compact. These zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics. Besides, an understanding of their statistical properties may prove useful in the analytical calculations of the wavefunctions in quantum chaotic systems. This motivates us to persue the present study which by a numerical statistical analysis shows that both long as well as the short range correlations exist between zeros; while the latter turn out to be universal and parametric-independent, the former seem to be system dependent and are significantly affected by various parameters i.e symmetry, localization etc. Furthermore, for the delocalized quantum dynamics, the distribution of these zeros seem to mimick that of the zeros of the random functions as well as random polynomials.Comment: PlainTex, Seven figures (available on direct request to author), J.Phys. (in Press

Eigenvalue Correlations For Banded Matrices

We study the evolution of the distribution of eigenvalues of $N\times N$ matrix ensembles subject to a change of variances of its matrix elements. Our results indicate that the evolution of the probability density is governed by a Fokker- Planck equation similar to the one governing the time-evolution of the particle- distribution in Wigner-Dyson gas, with relative variances now playing the role of time. This is also similar to the Fokker-Planck equation for the distribution of eigenvalues of a $N\times N$ matrix subject to a random perturbation taken from the standard Gaussian ensembles with perturbation-strength as the "time" variable. This equivalence alonwith the already known correlations of standard Gaussian ensembles can therefore help us to obtain the same for various physically-significant cases modeled by random banded Gaussian ensembles.Comment: Latex file, 5 pages, No figure

Level-Statistics of Disordered Systems: a Single Parametric Formulation

We find that the statistics of levels undergoing metal-insulator transition in systems with multi-parametric Gaussian disorders behaves in a way similar to that of the single parametric Brownian ensembles. The latter appear during aPoisson $\to$ Wigner-Dyson transition, driven by a random perturbation. The analogy provides the analytical evidence for the single parameter scaling behaviour in disordered systems as well as a tool to obtain the level-correlations at the critical point for a wide range of disorders.Comment: 4 Pages, 4 Figure

Random matrix ensembles with column/row constraints. II

We numerically analyze the random matrix ensembles of real-symmetric matrices with column/row constraints for many system conditions e.g. disorder type, matrix-size and basis-connectivity. The results reveal a rich behavior hidden beneath the spectral statistics and also confirm our analytical predictions, presented in part I of this paper, about the analogy of their spectral fluctuations with those of a critical Brownian ensemble which appears between Poisson and Gaussian orthogonal ensemble.Comment: This is the second part of the replaced version of the article reference arXiv:1409.6538v

Statistical analysis of chiral structured ensembles: role of matrix constraints

We numerically analyze the statistical properties of complex system with conditions subjecting the matrix elements to a set of specific constraints besides symmetry, resulting in various structures in their matrix representation. Our results reveal an important trend: while the spectral statistics is strongly sensitive to the number of independent matrix elements, the eigenfunction statistics seems to be affected only by their relative strengths. This is contrary to previously held belief of one to one relation between the statistics of the eigenfunctions and eigenvalues (e.g. associating Poisson statistics to the localized eigenfunctions and Wigner-Dyson statistics to delocalized ones).Comment: 23 pages (with double spacing), 7 figure

Criticality in the Quantum Kicked Rotor with a Smooth Potential

We investigate the possibility of an Anderson type transition in the quantum kicked rotor with a smooth potential due to dynamical localization of the wavefunctions. Our results show the typical characteristics of a critical behavior i.e multifractal eigenfunctions and a scale-invariant level-statistics at a critical kicking strength which classically corresponds to a mixed regime. This indicates the existence of a localization to delocalization transition in the quantum kicked rotor. Our study also reveals the possibility of other type of transitions in the quantum kicked rotor, with a kicking strength well within strongly chaotic regime. These transitions, driven by the breaking of exact symmetries e.g. time-reversal and parity, are similar to weak-localization transitions in disordered metals.Comment: 15 figures, to be published in Physical review E, 200

Improved Interference in Wireless Sensor Networks

Given a set ${\cal V}$ of $n$ sensor node distributed on a 2-dimensional plane and a source node $s \in {\cal V}$, the {\it interference problem} deals with assigning transmission range to each $v \in {\cal V}$ such that the members in ${\cal V}$ maintain connectivity predicate ${\cal P}$, and the maximum/total interference is minimum. We propose algorithm for both {\it minimizing maximum interference} and {\it minimizing total interference} of the networks. For minimizing maximum interference we present optimum solution with running time $O(({\cal P}_n + n^2) \log n)$ for connectivity predicate ${\cal P}$ like strong connectivity, broadcast ($s$ is the source), $k$-edge(vertex) connectivity, spanner, where $O({\cal P}_n)$ is the time complexity for checking the connectivity predicate ${\cal P}$. The running time of the previous best known solution was $O({\cal P}_n \times n^2)$ [Bil$\grave{o}$ and Proietti, 2008]. For the minimizing total interference we propose optimum algorithm for the connectivity predicate broadcast. The running time of the propose algorithm is O(n). For the same problem, the previous best known result was $2(1 + \ln (n-1))$-factor approximation algorithm [Bil$\grave{o}$ and Proietti, 2008]. We also propose a heuristic for minimizing total interference in the case of strongly connected predicate and compare our result with the best result available in the literature. Experimental results demonstrate that our heuristic outperform existing result.Comment: 10 pages, 1 figur