Transport in the random Kronig-Penney model

Gaussian fluctuations of products of random matrices distributed close to the identity

“…For example, the zero frequency limit of a harmonic chain is of this type, as well as the analysis of band centre and band edges of the Anderson model [7,8,5]. Also the behaviour near the so-called critical energies of a random Kronig-Penney model is of this perturbative nature [6]. It is the object of this work to develop a rigorously controlled perturbation theory for both the Lyapunov exponents and its variance within a generic model covering all the situations alluded to above, and potentially others.…”

“…In fact, the Lyapunov exponents in (i) and (iii) were calculated in [10] and [12] respectively, and the case of (ii) was sketched in [10,6], but there are actually considerable technical difficulties to make the analysis rigorous, see Section 4. On the other hand, no other work on variances at anomalies is known to us.…”

“…Both have already been considered in the work of Ishii [7], but there has been continuous and also recent interest in them (e.g. [1] and [6] respectively). The remainder of the article is mainly devoted to proofs.…”

“…The Lyapunov exponents were already obtained by the authors in the previous work [6], but the proofs and the error estimates there are given too briefly.…”

Pseudo-gaps for random hopping models

“…This drift is actually dictated by the Lyapunov exponent at E = 0: Histogram of Pruefer phases Figure 1: Schematic representation of the random dynamics as described in the text. The histogram shows the distribution of 10 5 Prüfer phases generated by (3) with a distributions of the hopping terms given by (6) where x is uniformly distributed in [−1, 1]. The parameters are c ev = 1.2, λ ev = 0.4, c od = 1 and λ od = 0 so that γ 0 < 0, and the energy E = 0.02.…”