Wishart稀疏随机矩阵乘积的谱密度。第一部分：稠密的情况
Spectral density of products of Wishart dilute random matrices. Part I: the dense case
摘要：
在这项工作中，我们研究了Wishart稀释的随机矩阵的形式$X(1)\cdots X(M)(X(1)\cdots X(M))^T$使用爱德华-琼斯技巧映射到一个系统的相互作用的粒子与随机耦合的多部图的产品的谱密度。在典型情况下，我们应用腔方法得到递归关系，从而得到谱密度。由于这个问题是相当丰富的，我们开始报告第一部分的情况下，密集矩阵的冗长的分析。本文导出了谱密度是一个$M+1$次多项式方程的解，并给出了$M=1$、$2$和$3$的精确表达式。对于一般的$M$，只有当所有的矩阵$X(t)$为$t=1,\ldots, M$时，我们才能找到谱密度的精确表达式。我们还对一般的$M$进行了一些观察，无可否认，这些观察是基于一些弱的数值证据，我们希望这些证据是正确的。
英文摘要：
In this work we study the spectral density of products of Wishart diluted random matrices of the form $X(1)\cdots X(M)(X(1)\cdots X(M))^T$ using the Edwards-Jones trick to map this problem into a system of interacting particles with random couplings on a multipartite graph. We apply the cavity method to obtain recursive relations in typical instances from which to obtain the spectral density. As this problem is fairly rich, we start by reporting in part I a lengthy analysis for the case of dense matrices. Here we derive that the spectral density is a solution of a polynomial equation of degree $M+1$ and obtain exact expressions of it for $M=1$, $2$ and $3$. For general $M$, we are able to find the exact expression of the spectral density only when all the matrices $X(t)$ for $t=1,\ldots, M$ are square. We also make some observations for general $M$, based admittedly on some weak numerical evidence, which we expect to be correct.
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