Delocalization in random polymer models

TL;DR

This paper investigates the spectral and transport properties of a class of one-dimensional random polymer models, revealing quadratic vanishing of the Lyapunov exponent at critical energies and demonstrating overdiffusive quantum transport despite localized eigenstates.

Contribution

It proves the quadratic vanishing of the Lyapunov exponent at generic critical energies and establishes optimal bounds on quantum transport in random polymer models.

Findings

Lyapunov exponent vanishes quadratically at critical energies

Density of states is positive at critical energies

Quantum transport is almost surely overdiffusive

Abstract

A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.