Extended states for the Random Schrödinger operator on $\mathbb{Z}^d$ ($d\geq 5$) with decaying Bernoulli potential

TL;DR

This work addresses delocalization for a discrete Schrödinger operator on $\mathbb{Z}^d$ with a decaying Bernoulli potential in dimensions $d\ge 5$, proving the existence of extended states for the regime $\tfrac{1}{4}<\alpha\le \tfrac{1}{3}$ and small coupling $\kappa$. The authors develop a 6th-order renormalization scheme, introduce a generalized Khintchine inequality via Bonami's lemma, and apply a fractional Gagliardo-Nirenberg inequality to manage new non-random operators arising at high order. A two-step resolvent rearrangement coupled with graph-analytic and decoupling techniques yields precise Green's function moment bounds and, ultimately, high-probability decay of the Green's function, together with the construction of non-decaying, bounded extended states. The results extend Bourgain’s prior bounds ($\alpha>1/3$) to a wider range and illuminate higher-dimensional mechanisms behind delocalization in decaying random environments, while highlighting technical obstacles that arise when attempting to push the threshold further toward $\alpha>0$. The approach blends random decoupling, hypercontractivity, and advanced renormalization to control intricate high-order interactions.

Abstract

In this paper, we investigate the delocalization property of the discrete Schrödinger operator $H_ω=-Δ+v_nω_nδ_{n,n'}$, where $v_n=κ|n|^{-α}$ and $ω=\{ω_n\}_{n\in\mathbb{Z}^d}\in \{\pm 1\}^{\mathbb{Z}^d}$ is a sequence of i.i.d. Bernoulli random variables. Under the assumptions of $d\geq 5$, $α>\frac14$ and $0<κ\ll1$, we construct the extended states for a deterministic renormalization of $H_ω$ for most $ω$. This extends the work of Bourgain [{\it Geometric Aspects of Functional Analysis}, LNM 1807: 70--98, 2003], where the case $α>\frac13$ was handled. Our proof is based on Green's function estimates via a $6$th-order renormalization scheme. Among the main new ingredients are the proof of a generalized Khintchine inequality via Bonami's lemma, and the application of the fractional Gagliardo-Nirenberg inequality to control a new type of non-random operators arising from the $6$th-order renormalization.

Figures (13)

• Figure 1: The characteristic graph of $G_0 v^6 G_0$.
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Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1: cf. Bon70ODO14
• proof
• proof
• Lemma 3.2