标题： 椭圆Ginibre系综的三个拓扑相与一个点电荷 显示英文标题

标题： Three topological phases of the elliptic Ginibre ensembles with a point charge

摘要： 我们考虑大小为$(c+1)N \times (c+1)N$的复数和辛椭圆 Ginibre 矩阵，这些矩阵被条件化为在$ p \in \mathbb{R} $处具有重数$ c N $的确定特征值。 我们证明它们的极限谱要么是单连通的，要么是双连通的，或者由两个不相连的单连通部分组成。 此外，记$\tau \in [0,1]$为非厄米参数，我们明确地刻画了参数空间$ (p, c, \tau) $中每种拓扑类型出现的区域。 对于滴状图是单连通或双连通的情况，我们提供了极限谱及其相应静电能量的显式描述。 作为应用，我们推导了在指数变化情形下椭圆 Ginibre 矩阵特征多项式的矩的渐近行为。 显示英文摘要

摘要： We consider the complex and symplectic elliptic Ginibre matrices of size $(c+1)N \times (c+1)N$, conditioned to have a deterministic eigenvalue at $ p \in \mathbb{R} $ with multiplicity $ c N $. We show that their limiting spectrum is either simply connected, doubly connected, or composed of two disjoint simply connected components. Moreover, denoting by $\tau \in [0,1]$ the non-Hermiticity parameter, we explicitly characterise the regions in the parameter space $ (p, c, \tau) $ where each topological type emerges. For cases where the droplet is either simply or doubly connected, we provide an explicit description of the limiting spectrum and the corresponding electrostatic energies. As an application, we derive the asymptotic behaviour of the moments of the characteristic polynomial for elliptic Ginibre matrices in the exponentially varying regime.