Delocalization of random band matrices in two and three dimensions · revision history

Revision 2

Accepted · core change · record

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Title
Delocalization of random band matrices in two and three dimensions
Plain
Random band matrices are simplified models of disordered conductors in which each site couples randomly to all sites within a distance W. The transition between trapped and spreading states in one dimension was settled in 2025, but in 2D and 3D at fixed band width it is open.
Precise · changed · core
Revision 1: Hermitian random band matrices on the box $\{1..L\}^{d}$ with variance profile of width $W$. Prove for $d = 3$ that bulk eigenvectors are delocalized for $W \ge W_0$ fixed and $L \to \infty$, and for $d = 2$ that the localization length scales as $\operatorname{exp}(c W^2)$. Answer: proofs of these scalings.
Revision 2: Hermitian random band matrices on the box $\{1..L\}^{d}$ with variance profile of width $W$. Prove for $d = 3$ that bulk eigenvectors are delocalized for $W \ge W_0$ fixed and $L \to \infty$, and for $d = 2$ that the localization length scales as $\exp(c W^2)$. Answer: proofs of these scalings.
Settled by · changed · core
Revision 1: Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\operatorname{exp}(c W^{2})$ localization length in $d = 2$.
Revision 2: Proofs of delocalization for fixed large $W$ in $d = 3$ and of the $\exp(c W^{2})$ localization length in $d = 2$.
Kind
well-posed
Answer type
proof
References
(empty)
Literature status
partially-resolved
Status note
In $1\mathrm{D}$ the threshold $W \sim L^{1/2}$ was established in 2025 (Yau and Yin, arXiv 2501.01718; Erdos and Riabov, arXiv 2506.06441; localization by Drogin, arXiv 2508.05802); in $2\mathrm{D}$ delocalization is proven for $W \ge L^{c}$, any $c > 0$ (Dubova, Yang, Yau and Yin, arXiv 2503.07606); fixed $W$ in $d = 2, 3$ remains open.
Posed since
(empty)

Decisions