# Adjacency matrices of random digraphs: singularity and anti-concentration

@article{Litvak2015AdjacencyMO, title={Adjacency matrices of random digraphs: singularity and anti-concentration}, author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef}, journal={arXiv: Probability}, year={2015} }

## 39 Citations

Invertibility of adjacency matrices for random d-regular directed graphs

- Mathematics
- 2018

Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random…

Invertibility of adjacency matrices for random d-regular graphs

- MathematicsDuke Mathematical Journal
- 2021

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*}…

Structure of eigenvectors of random regular digraphs

- MathematicsTransactions of the American Mathematical Society
- 2019

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a…

The smallest singular value of dense random regular digraphs

- Mathematics
- 2020

Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geq\lambda n$. We show that for any $\kappa \geq 0$,…

The circular law for random regular digraphs with random edge weights

- Mathematics
- 2015

We consider random $n\times n$ matrices of the form $Y_n=\frac1{\sqrt{d}}A_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with…

Circular law for sparse random regular digraphs

- Mathematics
- 2018

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.…

The smallest singular value of a shifted d-regular random square matrix

- MathematicsProbability Theory and Related Fields
- 2018

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…

The spectral gap of dense random regular graphs

- MathematicsThe Annals of Probability
- 2019

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular…

Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

- Mathematics
- 2018

We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we…

On sparse random combinatorial matrices

- Mathematics, Computer Science
- 2020

The proof accommodates sparse random combinatorial matrices in the sense that $d = o(n)$ is allowed and the singularity of deterministic integer matrices $A$ randomly perturbed by a sparse combinatorsial matrix is considered.

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