Random sorting networks: local statistics via random matrix laws
Random sorting networks: local statistics via random matrix laws
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DOI：
10.1007/s00440-018-0886-1
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发表时间：
2017-02
影响因子：
2
通讯作者：
V. Gorin;Mustazee Rahman
中科院分区：
文献类型：
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作者：
V. Gorin;Mustazee Rahman
本文发现了均匀随机排序网络交换过程的批量局部极限。极限对象是通过确定性过程定义的，这是 Edelman-Greene 算法的本地版本，应用于具有显式内核的二维行列式点过程。后者描述了反对称高斯酉系综中角点特征值接近 0 的渐近联合律。特别是，给定交换第一次出现在随机排序网络中的极限定律可以用反对称 GUE 中最接近 0 特征值的极限分布来确定。此外，给定交换的出现之间的渐近间隙总体上是高丁-梅塔定律——实对称随机矩阵的特征值之间的间隙的极限普遍分布。证明依赖于任意形状的随机泊松扬图核的行列式结构和双轮廓积分表示。
This paper finds the bulk local limit of the swap process of uniformly random sorting networks. The limit object is defined through a deterministic procedure, a local version of the Edelman–Greene algorithm, applied to a two dimensional determinantal point process with explicit kernel. The latter describes the asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric Gaussian Unitary Ensemble. In particular, the limiting law of the first time a given swap appears in a random sorting network is identified with the limiting distribution of the closest to 0 eigenvalue in the antisymmetric GUE. Moreover, the asymptotic gap, in the bulk, between appearances of a given swap is the Gaudin–Mehta law—the limiting universal distribution for gaps between eigenvalues of real symmetric random matrices. The proofs rely on the determinantal structure and a double contour integral representation for the kernel of random Poissonized Young tableaux of arbitrary shape.