通过RDT证明Lehner公式——不定与非对称场景
Proving Lehner's formulas via RDT -- indefinite and asymmetric scenarios
AI总结：
该研究通过发展随机对偶理论（RDT），在不定矩阵系数、非对称非平方变体等场景下证明Lehner公式，揭示其交错解耦性质，并结合半定规划验证了理论与数值结果的一致性。
AI中文摘要：
文献[28,53]建立的强渐近自由性允许通过对应的自由算子对应物研究高斯矩阵多项式的范数。对于对称克罗内克-高斯矩阵的半圆自由对应物的谱边，Lehner在文献[39]中确定了闭式解析表征。作为经典谱方法的替代，文献[62]中我们创建了基于随机对偶理论（Random Duality Theory, RDT）的框架来研究这些问题，并重新证明了矩阵系数为定常时的Lehner公式。在本工作中，我们进一步发展了RDT机制，并在几个关键方向取得了重要进展：(i) 对称平方情形：我们考虑不定矩阵系数，提供了与Lehner公式匹配的谱边下界；(ii) 非对称非平方变体：我们建立了Lehner公式的RDT非对称类似物，并证明它们是谱边的下界；(iii) 交错解耦性质：我们发现所得类似物具有显著的交错解耦性质。此外，我们考虑了半定规划（Semi-Definite Programming, SDP）公式，其可用于实际求解所得的解析表征。通过求解SDP得到的理论预测与数值模拟进行了比较，即使对于数百量级的问题维度，也显示出惊人的一致性。
英文摘要：
The strong asymptotic freeness established in [28,53] allows for the study of norms of Gaussian matrix polynomials via corresponding free operator counterparts. For spectral edges of semicircular free counterparts to symmetric Kronecker-Gaussian matrices, Lehner in [39] determined closed-form analytical characterizations. As an alternative to classical spectral methods, in [62], we created a Random Duality Theory (RDT) based framework for studying these problems and reproved Lehner's formula for definite matrix coefficients. In this work, we develop the RDT machinery further and achieve strong progress in several key directions: (i) Symmetric Square Case: We consider indefinite matrix coefficients and provide spectral edges lower bounds that match Lehner's formula. (ii) Asymmetric Non-Square Variants: We establish RDT asymmetric analogues to Lehner formulas and prove that they lower-bound the spectral edges. (iii) Interlaced Decoupling Property: We uncover that the obtained analogues exhibit a remarkable interlaced decoupling property. Additionally, we consider semi-definite programming (SDP) formulations that allow for the practical solution of the obtained analytical characterizations. Theoretical predictions obtained through solving SDPs are compared to numerical simulations, showing a striking agreement even for problem dimensions on the order of a few hundreds.
