发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非厄米随机带矩阵，在最优带宽阈值W_N≫log N及相关假设下，证明了周期模型等几类模型的圆律，完善了该领域的收敛性结果。
AI 中文摘要
我们考虑带宽随矩阵规模增长的非厄米随机带矩阵，研究其经验谱分布向圆律的收敛性。设N为矩阵规模，W_N为带宽。从普适性视角出发，学界推测只要W_N→∞，圆律就成立。此前的研究主要要求W_N≫N^{1/2}，唯一例外是文献[Han2511]，其针对开放边界的块三对角模型证明了W_N→∞时的圆律。本文针对几种周期模型，在近最优条件W_N≫log N下针对真正的离散模型证明了圆律。对于周期硬指示轮廓及其均匀、多项式缩放的推广形式，在有界密度和有限三阶矩假设下，只要W_N→∞，本文就证明了圆律。在同一阈值下，针对连续、可积且在每个有限区间上局部有下界的轮廓（包括指数衰减和高斯衰减轮廓，以及循环复高斯元素），本文也证明了圆律。对于周期全块模型，在有界密度和有限三阶矩假设下，只要W_N→∞，本文就证明了圆律。有界密度结果中的有限三阶矩假设可弱化为有限(2+α)阶矩假设。对于无密度假设的同一全块模型，在W_N≫log N时，针对中心化方差为1的实次高斯原子，本文证明了圆律。该证明使用了随机转移算子的复合：短辅助环上已确立的高带圆律输入校准了完整的外部系数范数，而边界均匀局部比较将该校准提升至目标环，即使N/W_N任意大时也适用。
英文摘要
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 $W_N\to\infty$. Previous results have mainly required $W_N\gg N^{1/2}$, with the principal exception of \cite{Han2511}, which proves the $W_N\to\infty$ 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 $W_N\gg\log N$ 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 $W_N\to\infty$. 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 $W_N\to\infty$. 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 $W_N\gg\log N$. 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.
Comments90 pages. Improved presentations. Lean formalisation statement included
