论文标题
有限的自由点过程
Finite Free Point Processes
论文作者
论文摘要
我们使用有限概率的技术来分析与特征值，奇异值和随机矩阵的广义奇异值相关的矩阵过程。我们使用的模型非常基本，分析完全由预期的特征多项式组成。我们的许多结果匹配随机矩阵理论的已知结果，但是我们的主要结果（关于广义奇异值）似乎比该领域中任何标准的随机矩阵过程（Hermite/laguerre/jacobi）更一般。为了测试这一点，我们对这个新过程进行了一系列模拟，一方面，这些过程证实了此过程可以表现出在标准的随机矩阵过程中看不到的行为，但另一方面，它提供了证据，证明我们的技术很好地捕获了真实的行为。这与我们能够计算出与标准模型相同的新模型的相同统计数据的事实表明，进一步的研究可能既有趣又富有成果。
We use techniques from finite free probability to analyze matrix processes related to eigenvalues, singular values, and generalized singular values of random matrices. The models we use are quite basic and the analysis consists entirely of expected characteristic polynomials. A number of our results match known results in random matrix theory, however our main result (regarding generalized singular values) seems to be more general than any of the standard random matrix processes (Hermite/Laguerre/Jacobi) in the field. To test this, we perform a series of simulations of this new process that, on the one hand, confirms that this process can exhibit behavior not seen in the standard random matrix processes, but on the other hand provides evidence that the true behavior is captured quite well by our techniques. This, coupled with the fact that we are able to compute the same statistics for this new model that we are for the standard models, suggests that further investigation could be both interesting and fruitful.
