Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box

TL;DR

The paper studies the discrete Anderson model on large boxes and introduces the principal eigenvalue $\lambda_{n,V}$ and the landscape function $L_{n,V}$ to understand low-lying eigenmodes. It formulates and sharpens a conjecture that the product $\lambda_{n,V}\|L_{n,V}\|_\infty$ converges almost surely to the constant $\frac{\mu_d}{2d}$ as the box size grows, with explicit scaling laws for $\lambda_{n,V}$ in two regimes (C1) and (C2) dictated by the potential distribution near zero. The main contributions include a rigorous upper bound for $\lambda_{n,V}$ via the largest low-potential ball and a matching lower bound via Lifshitz tails and IDS arguments, yielding a $\liminf$ result in general and a complete $\limsup$ in one dimension. The landscape-function analysis provides a robust framework with domain monotonicity and resolvent-identity tools, establishing that $\lambda_{n,V}\|L_{n,V}\|_\infty$ is universally bounded and converges to $\frac{\mu_d}{2d}$ in the general setting, with a complete 1D proof of the limsup. The work highlights a discrete-to-continuous constant transition (from $1+\frac{d}{4}$ to $\frac{\mu_d}{2d}$) and reinforces the landscape function as a predictor for extreme localization phenomena in random media.

Abstract

We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. We give a complete proof for the one dimensional case.

Figures (2)

• Figure 1: Empirical distribution of $\lambda_{n,V}\left(\frac{\omega_1\left| \ln F(0)\right|}{\ln n}\right)^{-2}-\mu_1$ for $d=1$ and $V(0)\overset{\text{d}}{=}$ Bernoulli$(0.3)$ computed form $10^5$ samples. The empirical mean ($m$) and empirical standard deviation ($s$) are shown in red and blue respectively.
• Figure 2: Empirical distribution of $\lambda_{n,V}\left\| L_{n,V}\right\|_\infty-\frac{\mu_1}{2}$ for $d=1$ and $V(0)\overset{\text{d}}{=}$ Bernoulli$(0.3)$ computed form $10^5$ samples. The empirical mean ($m$) and empirical standard deviation ($s$) are shown in red and blue respectively.

Theorems & Definitions (24)

• Conjecture 0
• Theorem 1
• Theorem 2
• Remark
• Proposition 3
• proof : Proof of Proposition \ref{['PrY']}
• Proposition 4
• proof
• Theorem 5: Theorem 1.3 of Biskup
• Remark