非对称向量戴森方程
Non-symmetric vector dyson equations
浏览论文内容
中文总结 AI 辅助
研究非对称向量戴森方程，建立其唯一向量解的测度分解与正则性。针对非对称矩阵\(S\)的奇异性和稳定性问题，开发图论方法，得出非回溯矩阵在不同情况的增长及稳定性估计，也得到周期设置下对称矩阵的相应估计。
中文摘要 AI 辅助
我们研究向量戴森方程\(-\frac{1}{m(z)} = z\mathbf{1}+\mathbf{a}+Sm(z)\)，其中参数\(z\)在复上半平面\(\mathbb{C}_+\)，\(\mathbf{a}\in\mathbb R^d\)且\(S\)是非负矩阵，不一定对称。此方程有唯一向量解\(m(z)\in\mathbb{C}_+^d\)，我们为此建立了完整的测度分解并证明了正则性。接着我们针对非对称矩阵\(S\)的奇异性和稳定性问题开发了一种图论方法。\(S\)的图结构确定了当\(z\)趋近实轴时稳定性算子可能的退化情况。特别地，对于非回溯矩阵，我们证明了在正则边处的平方根增长、在正则尖点处的立方根增长以及完整的稳定性估计。我们还在周期设置中获得了对称矩阵的相应估计。
英文摘要
We study the vector Dyson equation $$-\frac{1}{m(z)}=z\mathbf{1}+\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\mathbb{C}_+$, where $\mathbf{a}\in\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\in\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.
