arXiv 2106.005932021-06-22math.PR

稀疏矩阵：从无穷远看特征多项式的收敛性

Sparse matrices: convergence of the characteristic polynomial seen from infinity

Simon Coste

备注：: Added the semi-sparse case $d_n \to \infty$ and a more complete description of the $d_n=d<1$ case

摘要：

证明了具有iid $\mathrm{Bernoulli}(d/n)$元素的随机$n \times n$矩阵$A_n$的逆特征多项式$\det(I_n - zA_n)$按分布收敛于随机无穷乘积$\prod_{\ell = 1}^\infty(1-z^\ell)^{Y_\ell}$，其中$Y_\ell$是独立的$\mathrm{Poisson}(d^\ell/\ell)$.我们表明，这个随机函数是一个泊松模拟更经典的高斯对象，如高斯全纯混沌。作为一个副产品，我们得到了新的简单的证明以前的结果的渐近行为的稀疏Erdens-Rényi有向图：对于每一个$d>1$，最大的特征值的$A_n$是接近$d$和第二个最大的是小于$\sqrt{d}$，一个拉马努金性质的不规则有向图。对于$d<1$，$A_n$的唯一非零特征值收敛于单位圆上的Poisson多点过程。我们的结果也扩展到半稀疏制度，其中$d$被允许增长到$\infty$与$n$，比$n^{o(1)}$慢。我们发现，反向特征多项式收敛到一个更经典的对象写在对数相关的实高斯场的指数，在最近的论文\cite{bordenave2020convergence}研究的密集的情况下。在半稀疏的制度，$A_n/\sqrt{d_n}$的经验谱分布收敛到圆分布，作为我们的结果的结果，第二个特征值坚持圆的边缘。

英文摘要：

We prove that the reverse characteristic polynomial $\det(I_n - zA_n)$ of a random $n \times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d/n)$ entries converges in distribution towards the random infinite product $\prod_{\ell = 1}^\infty(1-z^\ell)^{Y_\ell}$ where $Y_\ell$ are independent $\mathrm{Poisson}(d^\ell/\ell)$ random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every $d>1$, the greatest eigenvalue of $A_n$ is close to $d$ and the second greatest is smaller than $\sqrt{d}$, a Ramanujan-like property for irregular digraphs. For $d<1$, the only non-zero eigenvalues of $A_n$ converge to a Poisson multipoint process on the unit circle. Our results also extend to the semi-sparse regime where $d$ is allowed to grow to $\infty$ with $n$, slower than $n^{o(1)}$. We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field, as in the dense case studied in a recent paper \cite{bordenave2020convergence}. In the semi-sparse regime, the empirical spectral distribution of $A_n/\sqrt{d_n}$ converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.

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摘要：

英文摘要：

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