Related papers: Absolutely Continuous Spectra of Quantum Tree Grap…
In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of…
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a…
Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees…
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…
In this paper we start a systematic study of quantum field theory on random trees. Using precise probability estimates on their Galton-Watson branches and a multiscale analysis, we establish the general power counting of averaged Feynman…
We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these…
Sparse trees are trees with sparse branchings. The Laplacian on some of these trees can be shown to have singular spectral measures. We focus on a simple family of sparse trees for which the dimensions can be naturally defined and shown to…
In this paper, we investigate spectral properties of discrete Laplacians. Our study is based on the Hardy inequality and the use of super-harmonic functions. We recover and improve lower bounds for the bottom of the spectrum and of the…
We show how certain properties of the Anderson model on a tree are related to the solutions of a non-linear integral equation. Whether the wave function is extended or localized, for example, corresponds to whether or not the equation has a…
We prove delocalization for the Anderson model on an infinite regular tree (or Cayley graph or Bethe lattice) at low disorder. This extends earlier results of Klein and Aizenman--Warzel by filling in the previously missing parts of the…
Following our work on the graph of the Weierstrass function, in the spirit of those of J. Kigami and R. S. Strichartz, which enabled us to build a Laplacian on the aforementioned graph, it was natural to go further and give the related…
We provide a simplified version of the geometric method given by Froese, Hasler and Spitzer and use it to prove the existence of absolutely continuous spectrum for a Cayley tree of arbitrary degree k.
We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under…
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…
We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on $\alpha$-stable trees, which lead in turn to short-time heat…
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then…
We investigate the relation between the eigenvalues of the Laplacian with Kirchhoff vertex conditions on a finite metric graph and a corresponding Titchmarsh-Weyl function (a parameter-dependent Neumann-to-Dirichlet map). We give a complete…
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…
For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $W\Gamma(R)$ has vertex set $Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in{\rm Ann}(x)$ and $s\in{\rm Ann}(y)$…
We consider standard subordinacy theory for general Sturm--Liouville operators and give criteria when boundedness of solutions implies that no subordinate solutions exist. As applications, we prove a Weidmann-type result for general…