Understanding the Random Displacement Model: From Ground-State Properties to Localization

TL;DR

This paper surveys recent advances in understanding the random displacement model, a quantum mechanical model of electrons in disordered media, highlighting results on ground states, spectral properties, and localization phenomena.

Contribution

It provides a comprehensive overview of recent results, including energy configurations, Lifshitz tail bounds, Wegner estimates, and proofs of localization at low energies.

Findings

Identification of minimal energy configurations

Lifshitz tail bounds on the density of states

Proof of spectral and dynamical localization at low energy

Abstract

We give a detailed survey of results obtained in the most recent half decade which led to a deeper understanding of the random displacement model, a model of a random Schr\"odinger operator which describes the quantum mechanics of an electron in a structurally disordered medium. These results started by identifying configurations which characterize minimal energy, then led to Lifshitz tail bounds on the integrated density of states as well as a Wegner estimate near the spectral minimum, which ultimately resulted in a proof of spectral and dynamical localization at low energy for the multi-dimensional random displacement model.