0000000000822710
AUTHOR
Nicole Tomczak-jaegermann
Circular law for sparse random regular digraphs
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 studyi…
Adjacency matrices of random digraphs: singularity and anti-concentration
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, …
The smallest singular value of a shifted $d$-regular random square matrix
We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$ and let $$\mathcal {M}_{n,d}$$ be the set of all $$n\times n$$ square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$ has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$ . Then the smallest singular value $$s_{n} (M)$$ of M is greater than $$n^{-6}$$ with probability at least $$1-C_2\log ^2 d/\sqrt{d}$$ , where c, $$C_1$$ , and $$C_2$$ are absolute positive constants independent of any other parameter…
Structure of eigenvectors of random regular digraphs
Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have cardin…
Anti-concentration property for random digraphs and invertibility of their adjacency matrices
Let Dn,dDn,d be the set of all directed d-regular graphs on n vertices. Let G be a graph chosen uniformly at random from Dn,dDn,d and M be its adjacency matrix. We show that M is invertible with probability at least View the MathML source1−Cln3d/d for C≤d≤cn/ln2nC≤d≤cn/ln2n, where c,Cc,C are positive absolute constants. To this end, we establish a few properties of directed d-regular graphs. One of them, a Littlewood–Offord-type anti-concentration property, is of independent interest: let J be a subset of vertices of G with |J|≤cn/d|J|≤cn/d. Let δiδi be the indicator of the event that the vertex i is connected to J and δ=(δ1,δ2,…,δn)∈{0,1}nδ=(δ1,δ2,…,δn)∈{0,1}n. Then δ is not concentrate…
The rank of random regular digraphs of constant degree
Abstract Let d be a (large) integer. Given n ≥ 2 d , let A n be the adjacency matrix of a random directed d -regular graph on n vertices, with the uniform distribution. We show that the rank of A n is at least n − 1 with probability going to one as n grows to infinity. The proof combines the well known method of simple switchings and a recent result of the authors on delocalization of eigenvectors of A n .