The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law
Abstract
We consider non-Hermitian random band matrices with growing bandwidth and study convergence of their empirical spectral distributions to the circular law. Let N denote the matrix size and WN the bandwidth. From a universality perspective, it is conjectured that the circular law holds whenever WN→∞. Previous results have mainly required WN≫ N1/2, with the principal exception of \cite{Han2511}, which proves the WN→∞ circular law for an open-boundary block-tridiagonal model. Here we prove the circular law at this optimal threshold for several periodic models and under the near-optimal condition WN≫log N for a genuinely discrete model. For the periodic hard-indicator profile and its uniform and polynomially tapered generalizations, we prove the circular law under bounded-density and finite-third-moment assumptions whenever WN→∞. At the same threshold, we prove the circular law for continuous, integrable profiles locally bounded below on every finite interval, including exponentially and Gaussian decaying profiles, with circular complex Gaussian entries. For the periodic full-block model, we prove the circular law under bounded-density and finite-third-moment assumptions whenever WN→∞. The finite-third-moment assumption in the bounded-density results can be weakened to a finite (2+α)-moment assumption. For the same full-block model without a density assumption, we prove the circular law for centered variance-one real subgaussian atoms when WN≫log N. The proof uses compositions of random transfer operators. An established high-band circular-law input on a short auxiliary ring calibrates the full exterior coefficient norm, and a boundary-uniform local comparison lifts this calibration to target rings even when N/WN is arbitrarily large.
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BibTeX
@misc{han2026circular,
title = {{The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law}},
author = {Yi Han},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01295},
eprint = {2609.01295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law
AU - Yi Han
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.01295
UR - https://arxiv.org/abs/2609.01295
AB - We consider non-Hermitian random band matrices with growing bandwidth and study convergence of their empirical spectral distributions to the circular law. Let N denote the matrix size and W_N the bandwidth. From a universality perspective, it is conjectured that the circular law holds whenever W_N→∞. Previous results have mainly required W_N≫ N^1/2, with the principal exception of \cite{Han2511}, which proves the W_N→∞ circular law for an open-boundary block-tridiagonal model. Here we prove the circular law at this optimal threshold for several periodic models and under the near-optimal condition W_N≫log N for a genuinely discrete model. For the periodic hard-indicator profile and its uniform and polynomially tapered generalizations, we prove the circular law under bounded-density and finite-third-moment assumptions whenever W_N→∞. At the same threshold, we prove the circular law for continuous, integrable profiles locally bounded below on every finite interval, including exponentially and Gaussian decaying profiles, with circular complex Gaussian entries. For the periodic full-block model, we prove the circular law under bounded-density and finite-third-moment assumptions whenever W_N→∞. The finite-third-moment assumption in the bounded-density results can be weakened to a finite (2+α)-moment assumption. For the same full-block model without a density assumption, we prove the circular law for centered variance-one real subgaussian atoms when W_N≫log N. The proof uses compositions of random transfer operators. An established high-band circular-law input on a short auxiliary ring calibrates the full exterior coefficient norm, and a boundary-uniform local comparison lifts this calibration to target rings even when N/W_N is arbitrarily large.
ER -