The Canopy Graph and Level Statistics for Random Operators on Trees

Abstract

For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended "canopy graph". For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular -pure point possibly with singular continuous component which is proven to occur in some cases.

Multifractality of wave functions on a Cayley tree: From root to leaves

“…The former result (the found behavior of high moments at s = 1) is consistent with the localization of eigenstates on canopy graphs that has been rigorously proven in Ref. 62.…”

“…Further, while our results imply the localized character of states at the boundary (which is in agreement with Ref. 62), we will show that certain moments of wave functions retain their multifractal behavior in this case as well.…”

“…Specifically, we have shown that the eigenfunction amplitudes at the root of the tree are distributed fractally in the most of the extended phase, which should be contrasted to the ergodicity of the extended phase in the RRG model. Interestingly, it is known on the mathematical level of rigor 62 that that random Schrödinger operators on socalled canopy graphs have pure-point spectrum for any strength of disorder, at least for some models of disorder distribution. A canopy graph is an infinite tree that represents a R → ∞ limit of a sequence of Cayley trees of "radius" R from the perspective of a boundary site.…”

Poisson Statistics of Eigenvalues in the Hierarchical Anderson Model

“…Minami's method has its origins in Molchanov's paper [26], where the first rigorous proof of the absence of energy level repulsion is given for a continuous one-dimensional model. After Minami's paper [25], the technique and its variations have been used to prove Poisson statistics of eigenvalues for different models [3,4,18,19,30]. In this paper, we combine existing and new results to prove Poisson statistics of eigenvalues for the hierarchical Anderson model (the precise definition of the model and the statement of our results are given in Section 3).…”

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

“…In this context, tree-like graphs are especially attractive for the analysis due to their recursive structure and the lack of circuits. In particular, regular trees (known as Cayley trees or Bethe lattices [6]) have become a standard trial template for various models of statistical physics (see, e.g., [1,2,3,34,37,61,63]), which are interesting in their own right but also provide useful insights into (harder) models in more realistic spaces (such as lattices Z d ) as their "infinite-dimensional" approximation [4,Chapter 4]. On the other hand, the use of Cayley trees is often motivated by the applications, such as information flows [38] and reconstruction algorithms on networks [15,36], DNA strands and Holliday junctions [48], evolution of genetic data and phylogenetics [14], bacterial growth and fire forest models [12], or computational complexity on graphs [18].…”