Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds
Abstract
We consider N × N random band matrices H = (Hxy) with centered complex Gaussian entries, indexed by points x,y on the two-dimensional discrete torus (ℤ / √N ℤ)2. The matrix entries Hxy vanish whenever the distance between x and y exceeds a constant multiple of the bandwidth parameter W. We prove that, for any fixed ε>0, if W≥(log N)2+ε, then all bulk eigenvectors are delocalized with high probability. Equivalently, this yields a lower bound of order exp(W1/2−ε) on the localization length of two-dimensional random band matrices for arbitrarily small constant ε>0, improving the superpolynomial lower bound WC for every fixed constant C>0 established in arXiv:2503.07606. A key ingredient of our proof is a new strategy that closes the resolvent flow analysis using only loop observables of order at most two, without relying on the higher-order loop hierarchy.
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BibTeX
@misc{lai2026localization,
title = {{Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds}},
author = {Xujie Lai and Fan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24426},
eprint = {2609.24426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24426},
}
RIS
TY - UNPB
TI - Localization Lengths for Two-Dimensional Random Band Matrices: Stretched-Exponential Lower Bounds
AU - Xujie Lai
AU - Fan Yang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.24426
UR - https://arxiv.org/abs/2609.24426
AB - We consider N × N random band matrices H = (H_xy) with centered complex Gaussian entries, indexed by points x,y on the two-dimensional discrete torus (ℤ / √N ℤ)^2. The matrix entries H_xy vanish whenever the distance between x and y exceeds a constant multiple of the bandwidth parameter W. We prove that, for any fixed ε>0, if W≥(log N)^2+ε, then all bulk eigenvectors are delocalized with high probability. Equivalently, this yields a lower bound of order exp(W^1/2−ε) on the localization length of two-dimensional random band matrices for arbitrarily small constant ε>0, improving the superpolynomial lower bound W^C for every fixed constant C>0 established in arXiv:2503.07606. A key ingredient of our proof is a new strategy that closes the resolvent flow analysis using only loop observables of order at most two, without relying on the higher-order loop hierarchy.
ER -