论文标题
Marchenko-Pastur分布的非热门概括：从循环法到多临界
A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
论文作者
论文摘要
我们考虑了WishArt类型的随机矩阵模型的复杂特征值$ x = x_1 x_2^*$，其中两个矩形复杂的Ginibre矩阵$ x_ {1,2} $ size $ n \ times（n+v N+v N+v n+v n+ν）$是通过非热门参数$ \ in [0,1] $进行的。对于一般$ν= o（n）$和$τ$，我们获得了全局限制密度及其支持，由移动的椭圆给出。它提供了Marchenko-Pastur分布的非热概括，当$τ= 1 $时，以最大相关为$ x_1 = x_2 $。复杂的Wishart特征值的平方根，对应于Dirac Matrix $ \ Mathcal {d} = \ begin {pmatrix} 0＆x_1 \\ x_2^*＆0 \ end end end end {pmatrix}的非零复杂的特征值。它以临界值$τ_c的形式显示了一个lemanistate类型的过渡，其中频谱的内部分为两个连接的组件。在多临界点，我们获得了平方变量中Ginibre合奏的边缘内核给出的局部核心。对于全球统计数据，我们将霜冻的平衡问题应用于2D库仑气体，而局部统计数据遵循复杂平面中正交Laguerre多项式核的鞍点分析。
We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+ν)$ are correlated through a non-Hermiticity parameter $τ\in[0,1]$. For general $ν=O(N)$ and $τ$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $τ=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $τ_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
