Related papers: Fractality of wave functions on a Cayley tree: Dif…
We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments $P_q=N \left<|\psi|^{2q}(i)\right>$ exhibit a…
We study bond percolation of the Cayley tree (CT) by focusing on the probability distribution function (PDF) of a local variable, namely, the size of the cluster including a selected vertex. Because the CT does not have a dominant bulk…
We review the state of the art on the delocalized non-ergodic regime of the Anderson model on Bethe lattices. We also present new results using Belief Propagation, which consists in solving the self-consistent recursion relations for the…
We investigate the random continuous trees called L\'evy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the…
The spectral statistics of a Cayley-tree is numerically studied. The statistics are non-universal due to the high ratio of boundary sites. Once the boundary sites are connected to each other in a way that preserves the local structure of…
We investigate the distribution of the resonances near spectral thresholds of Laplace operators on regular tree graphs with $k$-fold branching, $k \geq 1$, perturbed by nonself-adjoint exponentially decaying potentials. We establish results…
There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley…
We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrodinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior,…
In this work we introduce an energy function in order to study finite scale free graphs generated with different models. The energy distribution has a fractal pattern and presents log periodic oscillations for high energies. This…
We prove that every amenable one-ended Cayley graph has an invariant spanning tree of one end. More generally, for any 1-ended amenable unimodular random graph we construct a factor of iid percolation (jointly unimodular subgraph) that is…
In this paper we give a physically transparent picture of singular-continuous spectrum in disordered systems which possess a non-ergodic extended phase. We present a simple model of identically and independently distributed level spacing in…
Anderson localization on tree-like graphs such as the Bethe lattice, Cayley tree, or random regular graphs has attracted attention due to its apparent mathematical tractability, hypothesized connections to many-body localization, and the…
The decay of a quasiparticle in a confined geometry, resulting from electron-electron interactions, has been mapped onto the single-electron problem of diffusion on a Cayley tree by Altshuler et al. [Phys.Rev.Lett. 78, 2803 (1997)]. We…
The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian…
While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits…
Excitation waves are studied on trees and random networks of coupled active elements. Undamped propagation of such waves is observed in those networks. It represents an excursion from the resting state and a relaxation back to it for each…
In a deterministic or random tree, a notion of ancestral diversity can be defined as follows. Sample independently $n$ groups of $k$ leaves and count the number $N_n(k)$ of distinct most recent common ancestors of each of the groups. As $n$…
In this paper we study the evolution of the wave function with the system size in a locally periodic structure. In particular we analyse the dependence of the wave function with the number of unit cells, which also reflects information…
We study spectra of noncommutative dynamical systems, representations of fractal groups, and regular graphs. We explicitly compute these spectra for five examples of groups acting on rooted trees, and in three cases obtain totally…
Motivated from the study of eccentricity, center, and sum of eccentricities in graphs and trees, we introduce several new distance-based global and local functions based on the smallest distance from a vertex to some leaf (called the…