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

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.