论文标题
随机分类网络：边缘限制
Random sorting networks: edge limit
论文作者
论文摘要
排序网络是最短的路径，从$ 12 \ dots n $到$ n \ dots 21 $在对称组$ \ mathfrak s_n $的Cayley图中。本文将均匀随机分类网络的边缘本地限制计算为$ n \ to \ infty $。我们发现给定交换$（k，k+1）$首次出现的渐近分布，并通过$ 2K \ times 2k $ ague的最小正征值的定律来识别它（纯矩阵具有纯粹想象的高斯条目，它们独立于旋转 - 符合偏压的分布）。接下来，我们给出两个不同的间距的形式定义 - 在统一随机分类网络中给定交换$（k，k+1）$之间发生的时间距离。两个定义导致了两种不同的表达方式，这些表达方式是根据弗雷德姆决定因素的衍生物表达的。
A sorting network is a shortest path from $12\dots n$ to $n\dots 21$ in the Cayley graph of the symmetric group $\mathfrak S_n$ spanned by adjacent transpositions. The paper computes the edge local limit of the uniformly random sorting networks as $n\to\infty$. We find the asymptotic distribution of the first occurrence of a given swap $(k,k+1)$ and identify it with the law of the smallest positive eigenvalue of a $2k\times 2k$ aGUE (an aGUE matrix has purely imaginary Gaussian entries that are independently distributed subject to skew-symmetry). Next, we give two different formal definitions of a spacing -- the time distance between the occurrence of a given swap $(k,k+1)$ in a uniformly random sorting network. Two definitions lead to two different expressions for the asymptotic laws expressed in terms of derivatives of Fredholm determinants.
