论文标题
旋转不变二维库仑气体的数字差异的通用性
Universality of the number variance in rotational invariant two-dimensional Coulomb gases
论文作者
论文摘要
Lacroix-a-chez-toine，Majumdar和Schehr在[44]中建立了一个精确的地图，从Ginibre Ensemble中的复杂非热矩阵的$ n $复杂特征值与$ n $ n $ n $ n $ nterteracting fermions的位置的旋转陷阱的位置。一个重要的数量是fermions数量$ \ MATHCAL {n} _a $的统计信息。扩展了涵盖高斯和旋转不变电位$ q $的工作[44]，我们在平面复合物和符号合奏中进行了严格的分析，这两者都代表2D库仑气体。我们表明，当以平均密度与$ΔQ$成比例的单位测量时，$ \ MATHCAL {N} _a $的方差在很大程度上是通用的，而该单位本身是非宇宙的。当有限的分数或几乎所有费米子都在圆盘内部时，这在批量和边缘的限制很大。相比之下，起源于几乎没有特征值时，决定了普遍性类别的是潜力的奇异性。我们提供了三个明确的例子，来自Mittag-Leffler集合，Ginibre矩阵的产物和截断的单一随机矩阵。我们的证据分别利用了基本决定性的PFAFFIAN点过程的可集成结构，并在有限的$ n $上以截断的力矩为单位的差异表示差异的简单表示。
An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of $N$ non-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions $\mathcal{N}_a$ in a disc of radius $a$. Extending the work [44] covering Gaussian and rotationally invariant potentials $Q$, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the variance of $\mathcal{N}_a$ is universal in the large-$N$ limit, when measured in units of the mean density proportional to $ΔQ$, which itself is non-universal. This holds in the large-$N$ limit in the bulk and at the edge, when a finite fraction or almost all Fermions are inside the disc. In contrast, at the origin, when few eigenvalues are contained, it is the singularity of the potential that determines the universality class. We present three explicit examples from the Mittag-Leffler ensemble, products of Ginibre matrices, and truncated unitary random matrices. Our proofs exploit the integrable structure of the underlying determinantal respectively Pfaffian point processes and a simple representation of the variance in terms of truncated moments at finite-$N$.
