论文标题
确定点过程的海森堡家族的局部数差异和超均匀性
Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes
论文作者
论文摘要
复杂平面$ {\ mathbb {c}} $上特征值分布的批量缩放限制提供了一个确定点过程（DPP）。该点过程是无序的超一样系统的典型例子，其特征是大规模密度波动异常抑制。作为Ginibre DPP的扩展，我们考虑了在$ d $ d $二维复杂空间上定义的DPP家族，$ {\ Mathbb {c}} $，$ d \ in {\ Mathbb {n}} $，其中$ dpp在$ d = 1 = 1 $时实现了ginibre dpp。 DPPS的这个单参数系列（$ d \ in {\ Mathbb {n}} $）称为Heisenberg家族，因为与Heisenberg Group的Szegő内核一起确定了相关核。对于每个$ d $，使用修改后的贝塞尔功能，显示了一个精确而有用的表达式，显示了$ {\ mathbb {r}}^{2d} \ simeq {\ simeq {\ mathbb {c}}}^d $ in Ball in call in Ball的局部数字差异。我们证明，海森伯格家族中的任何DPP处于I类超均匀状态，从某种意义上说，数字差异的行为为$ r^{2d-1} $作为$ r \ to \ to \ infty $。我们的确切结果提供了大$ r $的数字差异的渐近扩展。
The bulk scaling limit of eigenvalue distribution on the complex plane ${\mathbb{C}}$ of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the $D$-dimensional complex spaces ${\mathbb{C}}$, $D \in {\mathbb{N}}$, in which the Ginibre DPP is realized when $D=1$. This one-parameter family ($D \in {\mathbb{N}}$) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each $D$, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius $R$ in ${\mathbb{R}}^{2D} \simeq {\mathbb{C}}^D$. We prove that any DPP in the Heisenberg family is in the hyperuniform state of Class I, in the sense that the number variance behaves as $R^{2D-1}$ as $R \to \infty$. Our exact results provide asymptotic expansions of the number variances in large $R$.
