论文标题
不变随机矩阵总和或乘积的顶部特征值的右大偏差原理
Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices
论文作者
论文摘要
在本说明中，我们研究了两个随机矩阵$ \ mathbf {a} $和$ \ mathbf {b} $的总特征值（或单数值）的正确大偏差，因为它们的尺寸为无限。矩阵$ \ mathbf {a} $和$ \ mathbf {b} $都被假定是从不变（或二式）合奏中取出的，具有限制电势，具有可能的\ emph {wall}，不允许使用eigenvalues/奇异值。这堵墙的引入将不同的模型置于非常通用的框架中。特别是，从大偏差的角度到考虑固定的对角线矩阵，壁恰好位于极限光谱密度的右边缘的情况是等效的。 \ cite {guionnetmaida20}。我们表明，参考文献中引入的倾斜方法。 \ cite {guionnetmaida20}可以扩展到我们的一般环境，相当于对特定操作的球形自旋玻璃模型的研究 - 对称矩阵 /对称矩阵的乘积 /矩形矩阵总和 - 我们正在考虑。
In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices $\mathbf{A}$ and $\mathbf{B}$ as their dimensions goes to infinity. The matrices $\mathbf{A}$ and $\mathbf{B}$ are each assumed to be taken from an invariant (or bi-invariant) ensemble with a confining potential with a possible \emph{wall} beyond which no eigenvalues/singular values are allowed. The introduction of this wall puts different models in a very generic framework. In particular, the case where the wall is exactly at the right edge of the limiting spectral density is equivalent, from the point of view of large deviations, to considering a fixed diagonal matrices, as studied previously in Ref. \cite{GuionnetMaida20}. We show that that the tilting method introduced in Ref. \cite{GuionnetMaida20} can be extended to our general setting and is equivalent to the study of a spherical spin glass model specific to the operation - sum of symmetric matrices / product of symmetric matrices / sum of rectangular matrices - we are considering.
