论文标题
广义稀释的Wishart集合的光谱特性
Spectral properties of the generalized diluted Wishart ensemble
论文作者
论文摘要
著名的Marčenko-Pastur Law认为随机协方差矩阵的渐近光谱密度已在物理，生物学，经济学，工程等方面发现了大量应用。在这里，使用自旋玻璃统计力学的技术，我们得出了有关广义稀释的Wishart矩阵光谱密度的简单公式。这些被定义为$ \ bm {f} \ equiv \ frac {1} {2d} \ left（\ bm {x} \ bm {y}^t+ \ bm {y} \ bm {y} \ bm {y} \ bm {x}^t \ right） P $矩形矩阵，其条目对应于分布后的双重加权随机双分式泊松图的链接$ p（x_ {i}^μ，y_ {i}^μ）= \ frac {d} {n} {n} \ varrho（x_ {i}^μ，y_ {i}^μ） +\左（1- \ frac {d} {n} \ right）Δ_{x_ {x_ {i}^μ，0}Δ__{y_ {y_ {i}^μ，0} $，使用概率密度$ \ varrho（x，y）$控制$ \ bm {x} $和$ \ bm {y} $之间的矩阵条目之间的相关性。我们的结果通过改变矩阵集合的参数，即图$ d $，矩阵$α= n/p $的矩形以及通过密度$ \ varrho（x，y）$的相关程度。最后，我们将我们的发现与数值对角线化进行了比较，显示出极好的一致性。
The celebrated Marčenko-Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as $\bm{F}\equiv \frac{1}{2d}\left( \bm{X}\bm{Y}^T+ \bm{Y}\bm{X}^T\right)$, where $\bm{X}$ and $\bm{Y}$ are diluted $N\times P$ rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution $P(x_{i}^μ,y_{i}^μ)=\frac{d}{N}\varrho(x_{i}^μ,y_{i}^μ)+\left(1-\frac{d}{N}\right)δ_{x_{i}^μ,0}δ_{y_{i}^μ,0}$, with the probability density $\varrho(x,y)$ controlling the correlation between the matrices entries of $\bm{X}$ and $\bm{Y}$. Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph $d$, the rectangularity of the matrices $α=N/P$, and the degree of correlation of the matrix entries via the density $\varrho(x,y)$. Finally, we compare our findings to numerical diagonalisation showing excellent agreement.
