正规矩阵的淬火增长与调和测度
Quenched Growth of Normal Matrices and Harmonic Measure
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中文总结 AI 辅助
本文研究大型正规随机矩阵特征值（二维库仑气体）在冻结部分特征值后继续采样时的增长行为，证明新特征值分布收敛于旧边界上的调和测度，并给出有限秩恒等式联系条件均值与冻结谱波动。
中文摘要 AI 辅助
大型正规随机矩阵的特征值构成二维库仑气体。在固定权重下，谱液滴随矩阵秩增长，其半经典密度增量为调和测度。我们冻结一个$N$特征值构型，并利用次数小于$N+M$且在冻结点处消失的多项式投影来采样另外$M$个特征值。它们的条件分布是冻结电荷势下的精确多项式系综。对于正则解析液滴，我们证明当$M/\log N\to\infty$且$M/N\to0$时，新特征值的空间分布收敛到旧边界上的调和测度。该收敛在冻结谱条件下成立，除了一组概率趋于零的冻结构型。由次数为$N,\ldots,N+M-1$的正交多项式形成的平衡过程具有相同的极限。一个精确的有限秩恒等式将条件均值的多项式矩与冻结谱的波动联系起来，其余项来自次数为$N+M$及以上的多项式模式。
英文摘要
The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an $N$-eigenvalue configuration and sample $M$ further eigenvalues using the projection onto polynomials of degree less than $N+M$ that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when $M/\log N\to\infty$ and $M/N\to0$. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees $N,\ldots,N+M-1$ has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree $N+M$ and above.
发表机构
- Saint Petersburg State University（圣彼得堡国立大学）
- HSE University (National Research University Higher School of Economics)（高等经济大学）
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