Poisson Statistics of Eigenvalues in the Hierarchical Anderson Model

Abstract

We study the eigenvalue statistics for the hieracharchial Anderson model of Molchanov [21][22][23]27,28]. We prove Poisson fluctuations at arbitrary disorder, when the the model has a spectral dimension d < 1. The proof is based on Minami's technique [25] and we give an elementary exposition of the probabilistic arguments.

Low-temperature dynamics of long-ranged spin-glasses: full hierarchy of relaxation times via real-space renormalization

“…The Dyson hierarchical ferromagnetic Ising model [60] has been much studied by both mathematicians [61][62][63][64] and physicists [65][66][67][68][69]. More recently, various Dyson hierarchical versions of disordered systems have been considered, in particular Anderson localization models [72][73][74][75][76][77][78][79], random fields Ising models [70,71] and spin-glasses [80][81][82][83][84].…”

Renormalization flow of the hierarchical Anderson model at weak disorder

“…Therefore, close to the upper spectral edge E pure ∞ , the integrated DOS exhibits the asymptotic behaviour 8,12,14,15 …”

“…Contrary to the rigorous results established for the density of states (DOS), 8,[12][13][14][15] less work has been devoted to the study of the nature of the eigenstates of the HAM. In a recent paper, 16 the authors have shown the existence of an extended phase for a sufficiently slow decay of the hopping energy, in contrast to a previous conjecture stating that all states should be localized.…”

Renormalization Group Analysis of the Hierarchical Anderson Model

“…As mentioned before, the first proof of complete spectral localization for the hierarchical model at any c > 0 in the special case of a Cauchy probability density is due to S. Molchanov [26]. Later, E. Kritchevski showed [18,20] that H almost surely has only pure-point spectrum provided that either V has a Cauchy component or c > 1/2 and that the rescaled eigenvalues converge to a Poisson point process if c > 2. Proposition 2.2 gets rid of any conditions on the distribution or the spectral dimension and may be considered optimal as far as spectral localization is concerned.…”

“…The first proof of complete spectral localization for the hierarchical model in the special case of a Cauchy random potential of arbitrary strength is due to S. Molchanov [26]. His proof ideas where later extended to more general distributions by E. Kritchevski [18,20]. Notably, neither of these works proved or even conjectured the appearance of a delocalized phase in the regime of longrange, but summable hopping strength of the hierarchical Laplacian.…”