Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.Comment: 8 pages, 6 figure

A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $\epsilon$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $\epsilon$, $P(\epsilon)\sim 1/\epsilon^{1+\alpha}$ with $\alpha\in(0,2)$. Our model serves as a generalization of 1D Lloyd's model, which corresponds to $\alpha=1$. First, we verify that the ensemble average $\left\langle -\ln G\right\rangle$ is proportional to the length of the wire $L$ for all values of $\alpha$, providing the localization length $\xi$ from $\left\langle-\ln G\right\rangle=2L/\xi$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $\alpha$ and $\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$ is proportional to $G^\beta$, for $G\to 0$, with $\beta\le\alpha/2$, in agreement to previous studies.Comment: 5 pages, 5 figure

Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action \left as a function of the $n$-th iteration of the map as well as the parameters $K$ and $\gamma$, controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., $K\gg 1$. In this regime and for large initial action $I_0\gg K$, we prove that dissipation produces an exponential decay for the average action \left. Also, for $I_0\cong 0$, we describe the behavior of \left using a scaling function and analytically obtain critical exponents which are used to overlap different curves of \left onto an universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action $\omega$.Comment: 20 pages, 7 figure