# Circular law for sparse random regular digraphs

@article{Litvak2018CircularLF, title={Circular law for sparse random regular digraphs}, author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef}, journal={arXiv: Probability}, year={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. 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…

## 16 Citations

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 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…

On the second eigenvalue of random bipartite biregular graphs

- Mathematics
- 2020

We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the…

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…

The rank of random regular digraphs of constant degree

- Mathematics, Computer ScienceJ. Complex.
- 2018

It is shown that A_n has rank at least at least $n-1$ with probability going to one as $n$ goes to infinity.

The circular law for sparse non-Hermitian matrices

- MathematicsThe Annals of Probability
- 2019

For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and…

Singularity of sparse Bernoulli matrices

- Mathematics
- 2020

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,…

Singularity of Bernoulli matrices in the sparse regime $pn = O(\log(n))$

- Mathematics
- 2020

Consider an $n\times n$ random matrix $A_n$ with i.i.d Bernoulli($p$) entries. In a recent result of Litvak-Tikhomirov, they proved the conjecture $$ \mathbb{P}\{\mbox{$A_n$ is singular}\}=(1+o_n(1))…

The sparse circular law under minimal assumptions

- MathematicsGeometric and Functional Analysis
- 2019

The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $${n \times n}$$n×n matrix with i.i.d. entries converges to the uniform measure on the unit disc as…

Spectral gap of sparse bistochastic matrices with exchangeable rows

- Mathematics, Physics
- 2018

We consider a random bistochastic matrix of size $n$ of the form $M Q$ where $M$ is a uniformly distributed permutation matrix and $Q$ is a given bistochastic matrix. Under mild sparsity and…

## References

SHOWING 1-10 OF 47 REFERENCES

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…

On the singularity of adjacency matrices for random regular digraphs

- Mathematics
- 2014

We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming $$\min (d,n-d)\ge C\log…

The Circular Law for random regular digraphs

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2019

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends…

Adjacency matrices of random digraphs: singularity and anti-concentration

- Mathematics
- 2015

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…

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…

Circular law for the sum of random permutation matrices

- Mathematics
- 2017

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le…

Local Kesten–McKay Law for Random Regular Graphs

- Mathematics, PhysicsCommunications in Mathematical Physics
- 2019

We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum $${[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]}$$[-2d-1+ε,2d-1-ε] down to…

The circular law for sparse non-Hermitian matrices

- MathematicsThe Annals of Probability
- 2019

For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and…

Invertibility of Sparse non-Hermitian matrices

- Mathematics
- 2015

We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are…