How large is large? Estimating the critical disorder for the Anderson model

TL;DR

This paper establishes the precise large disorder threshold for complete localization in the d-dimensional Anderson model, connecting it to the self-avoiding walk connective constant and confirming a longstanding conjecture by Anderson.

Contribution

The authors derive an exact formula for the critical disorder strength in the Anderson model, linking it to lattice properties and confirming Anderson's original proposal.

Findings

Complete localization occurs for disorder strength above the threshold

The threshold is given by a specific equation involving the self-avoiding walk constant

The result confirms Anderson's 1958 conjecture about the large disorder limit

Abstract

Complete localization is shown to hold for the $d$-dimensional Anderson model with uniformly distributed random potentials provided the disorder strength $\lambda >\lambda_{And}$ where $\lambda_{\text{And}}$ satisfies $\lambda_{\text{And}}=\mu_d e \ln \lambda_{\text{And}}$ with $\mu_d$ the self-avoiding walk connective constant for the lattice $\mathbb{Z}^d$. Notably $\lambda_{\text{And}}$ is precisely the large disorder threshold proposed by Anderson in 1958.