Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.

中文翻译：

---

稀疏随机正则有向图的循环律

固定一个常量 $C\geq 1$ 并让 $d=d(n)$ 满足 $d\leq \ln^{C} n$ 对于每个大整数 $n$。用$A_n$表示$n$顶点上的均匀随机有向$d$-正则图的邻接矩阵。我们表明，只要 $d\to\infty$ 与 $n$，适当重新缩放的矩阵 $A_n$ 的经验谱分布在概率上会弱收敛于循环定律。这一结果与库克的早期工作一起，彻底解决了度数趋于无穷大的有向 $d$-正则设置中经验分布弱收敛的问题。作为我们证明的一个关键要素，我们基于研究行空间的随机法线和构建乘积结构来处理矩阵条目之间缺乏独立性的问题，开发了一种限制 $A_n$ 中间奇异值的技术。