In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j<k\leq N}(x_j-x_k)^2\prod_{j=1}^N (1+ix_j)^{-s-N}(1-ix_j)^{-\bar{s}-N}dx_j,$$where $s$ is a complex number such that $\Re(s)>-1/2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $\Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1/N)$.
On the Largest Singular Values of Random Matrices with Independent Cauchy Entries
利用行列式方法研究了具有独立柯西元的大型$ m \times n $实矩形随机矩阵的最大奇异值的分布。我们表明，统计性质（重新缩放的$ \frac{1}{m^2\*n^2}$的一个因素）最大的奇异值同意在极限与强度$ \frac{1}π x^{-3/2} $的非齐次泊松随机点过程的统计，因此，是不同的Tracy-Widom定律。在我们的方法的其他推论，我们显示了一个有趣的数学期望之间的联系复杂的矩形$ m \times n $标准Wishart合奏和真正的矩形$ 2m \times 2n $标准Wishart合奏的决定因素。
Cluster tails for critical power-law inhomogeneous random graphs
最近，得到了具有有限方差但无限三阶矩度的rank-1型临界非齐次随机图的簇大小的标度极限（参见Bhamidi，van der Hofstad和van Leeuwaarden以前的工作）。证明了当度服从指数在区间（3，4）内的幂律时，按大小递减排序并适当缩放的簇序列随着n的无穷大收敛于一个递减的非退化随机变量序列.在这里，我们研究了重新缩放的最大集群的极限的尾部，即，作为u的函数，最大簇的缩放限制取大值u的概率。这将Pittel关于Erdens-Rényi随机图的一个相关结果推广到了秩为1且三阶矩度为无穷大的非齐次随机图的情形.我们利用微妙的大偏差和弱收敛的论点。
