equilibrium state is extended both with respect to its gyration radius and to its Lyapunov exponent. In addition, we show that the phenomenon of dynamical localization is only marginally connected with the spectrum of the transition rate matrix, and is dominated by the properties of its eigenfunctions which differ significantly in dimensions one and three.'
author:
- 'Franziska Flegel and Igor M. Sokolov'
title: Dynamical localization and eigenstate localization in trap models
---

Introduction
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The random trap model together with its close relative, the barrier model, can be applied to a variety of physical problems [@Haus1987] which are related to random walks in disordered media, like properties of a photocurrent in amorphous solids [@Scher1975], or have an equivalent mathematical representation, like the behavior of resistor and capacitance networks [@Bouchaud1990] or lattice vibrations of harmonic chains [@Alexander1981; @Dyson1953]. It has also become prominent as a toy model for the phase space dynamics of spin glasses [@Bouchaud1992].

The random trap model on a regular lattice consists of a trapping landscape $\{ E_k \}$ of energy wells located on the lattice’s sites $k$. The $E_k$ are iid random variables which in our case have the common density $$\begin{aligned}
     \rho_E (\epsilon) &= \frac{1}{kT_g} 