Related papers: Diffusion, localization and dispersion relations o…

The spectral properties of the restricted fractional Laplacian with Dirichlet boundary conditions in a smoothly bent waveguide is investigated. The existence of eigenvalues below the threshold of the continuous spectrum is proved,…

In this lecture we review recent lattice QCD studies of the statistical properties of the eigenvalues of the QCD Dirac operator. We find that the fluctuations of the smallest Dirac eigenvalues are described by chiral Random Matrix Theories…

We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on $d$-dimensional lattices. The model may be recast as an…

We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the…

We study the properties of the ``Rigid Laplacian'' operator, that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the…

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief…

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the…

The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space.…

We propose a simplified version of the Multi-Scale Analysis of tight-binding Anderson models with strongly mixing random potentials which leads directly to uniform exponential bounds on decay of eigenfunctions in arbitrarily large finite…

Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths.It is well-known that higher clustering drives localization while shorter path length supports delocalization of…

One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gomez et al. [Phys. Rev. Lett. 101, 028701…

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions…

We study the properties of marginal distributions-projections of the phase space representation of a physical system-under relativistic transforms. We consider the Galileo case as well as the Lorentz transforms exploiting the relativistic…

The dispersion relation for magnetoplasmons in a planar superlattice with periodically alternating regions of gapless and gapped modications of graphene has been derived within the frame of the random-phase approximation. The contribution…

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying…

In most studies, street networks are considered as undirected graphs while one-way streets and their effect on shortest paths are usually ignored. Here, we first study the empirical effect of one-way streets in about $140$ cities in the…

In an attempt to characterize the structure of eigenvectors of random regular graphs, we investigate the correlations between the components of the eigenvectors associated to different vertices. In addition, we provide numerical…

We study the distribution and scaling of the extreme height fluctuations for Edwards-Wilkinson-type relaxation on small-world substrates. When random links are added to a one-dimensional lattice, the average size of the fluctuations becomes…

We analyze the effect of correlations in a simple model of small world network by obtaining exact analytical expressions for the distribution of shortest paths in the network. We enter correlations into a simple model with a distinguished…

We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations,…