Pseudo-gaps for random hopping models

Abstract

For one-dimensional random Schrödinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Prüfer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy, at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models. The main result is a Hölder continuity of the rotation number at the critical energy that, under certain conditions on the randomness, implies the existence of a pseudo-gap. The proof uses renewal theory. The result is illustrated by numerics.

Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents

“…with sequences (t ω (n)) n∈Z and (v ω (n)) n∈Z of compactly supported, positive and real random numbers, respectively, called the hopping and potential values. Here, we will focus on particular kinds of random Jacobi matrices, namely so-called random polymer models [10,8]. In these models, H ω is built from independently drawn blocks of random length K. Each such block is called a polymer and is given by the data σ = (K,t σ (1), .…”

“…Hence, σ ∈ Σ, where Σ is a compact subset of L K=1 {K} × R K + × R K equipped with a probability measure P Σ . How to construct the dynamical system Ω as a Palm measure is explained in detail in [10,8], but this is not relevant for the following. The best known example is the Anderson model in which K = 1 and t ω (n) = 1 and only the potential values are random and given by an i.i.d.…”

Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents

“…with (t ω (n)) n∈Z and (v ω (n)) n∈Z sequences of compactly supported, positive and real random numbers, respectively, called the hopping and potential values. Here, we will focus on particular kinds of random Jacobi matrices, namely so-called random polymer models [9,7]. In these models, H ω is built from independently drawn blocks of random length K. Each such block is called a polymer and is given by the data σ = (K, tσ (1), .…”

“…which is supposed to be compact and equipped with a probability measure P Σ . How to construct the dynamical system Ω as a Palm measure is explained in detail in [9,7], but this is not relevant for the following. The best known example is the Anderson model in which K = 1 and t ω (n) = 1 and only the potential values are random and given by an i.i.d.…”