论文标题
随机排列和常规挖掘的总和的特征多项式
The characteristic polynomial of sums of random permutations and regular digraphs
论文作者
论文摘要
令$ a_n $为$ d $置换矩阵的总和$ n \ times n $，每个矩阵均以随机和独立绘制。我们证明，归一化的特征多项式$ \ frac {1} {\ sqrt {d}} \ det（i_n -z a_n/\ sqrt {d}）$收敛时，$ n \ to \ n \ to \ n \ to \ infty $ clem to \ infty $ to \ infty $ to \ inpty $ for f to \ inpty $ for \ nifty $。作为应用程序，我们获得了随机常规挖掘的光谱间隙的基本证明。我们的结果在固定$ d $的政权中都是有效的，对于$ d $，$ n $ slagh。
Let $A_n$ be the sum of $d$ permutation matrices of size $n\times n$, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial $\frac{1}{\sqrt{d}}\det(I_n - z A_n/\sqrt{d})$ converges when $n\to \infty$ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$.
