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.

7 pages, latex; typos corrected and a more thorough discussion of the relation to previous work added

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