Hierarchical Anderson model

Renormalization Group Analysis of the Hierarchical Anderson Model

“…As an aside, we note that by extending an argument of E. Kritchevski [18], the spectral statement may also be established without Assumption 1.1.…”

“…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.…”

“…We will be particularly interested in the quantities G n (j, k; z) = δ k , (H ∞,n − z) −1 δ j g n (j; z) = 2 −n k∈Bn(j) δ k , (H ∞,n − z) −1 δ j Q n (j, z) = ϕ n (j), (H ∞,n − z) −1 ϕ n (j) with k ∈ B n (j) and ϕ n (j) = 2 −n/2 1 Bn(j) . Proposition 2.2 of [18] contains the formula G n (j, k; z) = G 0 (j, k; z) − n r=d(j,k)…”

“…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.…”

“…We hope that our present work will stimulate further research in this direction. It is possible to define hierarchical Anderson models corresponding to more general hierarchical structures than the one above, see [18,19], but since the analysis of such generalizations does not require any fundamentally new ideas we restrict ourselves to the present situation here.…”