AI 中文总结
研究强塞格极限定理的多元推广，用独立随机酉矩阵系统代替单个酉矩阵，经典证明不适用，引入新工具，还得到酉矩阵束平均行列式及自由哈尔酉矩阵矩阵和谱半径公式等结果。
AI 中文摘要
强塞格极限定理是关于大型托普利兹矩阵行列式渐近性的定理，可重新表述为关于随机酉矩阵特征值统计的概率陈述。我们以后一种形式证明了该定理的多元推广，用独立随机酉矩阵系统代替单个酉矩阵。经典定理的标准证明在此情形下不适用，需引入随机矩阵理论、自由概率和非交换函数理论的工具。此外，还得到了酉矩阵规模趋于无穷时随机酉矩阵束平均行列式的结果，以及自由哈尔酉矩阵矩阵和谱半径公式的辅助结果。
英文摘要
The Strong Szegő Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. It can be reformulated as a probabilistic statement about eigenvalue statistics of random unitary matrices. We prove a multivariate generalization of the theorem in this latter form, replacing a single unitary with a system of independent random unitaries. It turns out that the standard proofs of the classical theorem do not generalize to this setting, and instead we must import tools from random matrix theory, free probability, and noncommutative function theory. In addition, we obtain results about the averaged determinants of random unitary pencils, as the size of the unitaries tends to infinity; and an auxiliary result giving a formula for the spectral radius of a matricial sum of free Haar unitaries.
