Resonances and Partial Delocalization on the Complete Graph

Abstract

Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schr\"odinger operator on the complete graph. The operators exhibits local quasi modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the $\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the statistics of \v{S}eba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz-Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes

On the ubiquity of the Cauchy distribution in spectral problems

“…Furthermore, combined with (6.16), the arguments imply that the joint distribution of the pair of random variables F (1) ω,n (x + i0), F (2) ω,n (x) is continuous in the limit, and hence (6.9) holds. Theorem 6.2 has implications for the random matrix models which are discussed next, and for thě Seba process [27,23,6,3] on which more is said in [2]. Following is another continuity criterion which may be of interest beyond the cases covered by it, in particular when the spectral measures are singular but with dense support and not of uniform masses.…”

“…Thus, the assumed convergence of the spectral measure allows to deduce the continuity of the probability distribution of F (1) ω,n (x + i0) for sites x ∈ [−1/2, 1/2] at which (6.10) holds. The probability distribution of F (2) ω,n (x + i0) is continuous in the limit n → ∞ by an application of Lemma 5.2. Furthermore, combined with (6.16), the arguments imply that the joint distribution of the pair of random variables F (1) ω,n (x + i0), F (2) ω,n (x) is continuous in the limit, and hence (6.9) holds.…”

“…The probability distribution of F (2) ω,n (x + i0) is continuous in the limit n → ∞ by an application of Lemma 5.2. Furthermore, combined with (6.16), the arguments imply that the joint distribution of the pair of random variables F (1) ω,n (x + i0), F (2) ω,n (x) is continuous in the limit, and hence (6.9) holds. Theorem 6.2 has implications for the random matrix models which are discussed next, and for thě Seba process [27,23,6,3] on which more is said in [2].…”

“…Of particular interest here are the scaling limits in which spectra of finite dimensional operators of increasing dimension are studied on a scale of the eigenvalue spacing [27,4,7,6,23,18,2]. The functions of interest may be found to converge in an appropriate distributional sense to random HP functions of singular spectrum which in the simplest case consists of simple poles located along R. In the latter case, (1.2) extends to a random meromorphic functions, whose spectra form a random point process on R.…”

Renormalization Group Analysis of the Hierarchical Anderson Model

“…This belief was recently challenged by F. Metz, L. Leuzzi, G. Parisi, and V. Sacksteder [23], who reported numerical evidence for the appearance of a special energy in (effective spectral) dimension d > 2, for which delocalized eigenvectors appear in the case of a weak Gaussian random potential. Superficially, this numerical result looks related to the existence of resonant delocalization at special energies in yet another toy version of the Anderson model, namely that on the complete graph [1]. One of the main aims of this paper is to argue that as long as the hierarchical hopping is summable, all states are localized in every possible specification of that term and the eigenvalues exhibit Poisson statistics.…”