Olivier Bourget, Diomba Sambou, Amal Taarabt
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 on the absence of resonances which in particular involve absence of discrete spectrum near some sectors of the essential spectrum of the operators.
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The Schrödinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an orthogonal decomposition of the space L_2 on the tree which reduces the Schrödinger operator with any symmetric weight. Using this decomposition, we analyse the spectrum of such operators, including the free Laplacian, under various assumptions about the tree and the potential.
Spectral properties of the Schroedinger operator $A_λ = -Δ+λV$ on regular metric trees are studied. It is shown that as $λ$ goes to zero the behavior of the negative eigenvalues of $A_λ$ depends on the global structure of the tree.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
A metric tree is a tree whose edges are viewed as line segments of positive length. The Dirac operator on such tree is the operator which operates on each edge, complemented by the matching conditions at the vertices which were given by Bolte and Harrison \cite{BolteHarrison2003}. The spectrum of Dirac operator can be quite different, reflecting geometry of the tree. We discuss a special case of trees, namely the so-called regular trees. They possess a rich group of symmetries. This allows one to construct an orthogonal decomposition of the space $L^2(Γ)$ which reduces the Dirac. Based upon th…
We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to $-\infty$, a narrow cluster of finitely many eigenvalues tends to $-\infty$, while the eigenvalues above the cluster remain bounded from below. Certain "rogue" eigenvalues break away from this cluster and tend even faster toward $-\infty$. The spectrum can be visualized as the intersection points of two objects in the plane--a spiral curve depending on the Schrödinger potential, and a set o…
We prove dispersive estimates for two models~: the adjacency matrix on a discrete regular tree, and the Schrödinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schrödinger operators on the real line. We establish a $t^{-3/2}$-decay for both models which is sharp, as we give the first-order asymptotics.
The Arboreal gas model on a finite graph $G$ is the Bernoulli bond percolation on $G$ conditioned on the event that the sampled subgraph is a forest. In this short note we study the arboreal gas on a regular tree wired at the leaves and obtain a comprehensive description of the weak limit of this model.
We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for …
We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of such operators is proven to be purely absolutely continuous and to consist of finitely many intervals. We further investigate stability of the absolutely continuous spectrum under perturbations by sufficiently small potentials. On the one hand, we look at a class of deterministic potentials which include radial symmetric ones. The absolutely continuous spec…
Quantum percolation conjecture on regular trees
本文证明在度数至少为3的正则树上，伯努利键逾渗中无限簇的出现严格早于绝对连续谱，结合Bordenave定理得到$p_c<p_{ac}<1$。
We consider the family of real (generalized) eigenfunctions of the adjacency operator on $T_d$ - the $d$-regular tree. We show the existence of a unique invariant Gaussian process on the ensemble and derive explicitly its covariance operator. We investigate the typical structure of level sets of the process. In particular we show that the entropic repulsion of the level sets is uniformly bounded and prove the existence of a critical threshold, above which the level sets are all of finite cardinality and below it an infinite component appears almost surely.
The regular tree Anderson model at low disorder
该研究证明了无限正则树安德森模型在低无序度下的离域性，填补谱缺失部分以扩展早期成果，论证可推广至特定无序情况，并证明无序度消失时李雅普诺夫指数的连续性。
We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.
We consider noisy binary channels on regular trees and introduce periodic enhancements consisting of locally self-correcting the signal in blocks without break of the symmetry of the model. We focus on the realistic class of within-descent self-correction realized by identifying all descendants $k$ generations down a vertex with their majority. We show that this also allows reconstruction strictly beyond the critical distortion. We further identify the limit at which the critical distortions of within-descent $k$ self-corrected transmission converge, which turns out to be the critical point fo…
We present a proof of the power law decay of magnetic moment for the $q$-state antiferromagnetic Potts model on the regular tree at the critical temperature, and also justify that the exact exponent is $\frac{1}{2}$. Our proof relies on the assumption of the uniqueness at the critical temperature, which has been established for $q=3,4$, and for $q \ge 5$ with large degree. An iterative contraction inequality is developed for independent interests.
We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is $\mathbb Z$. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approx…
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