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
• Mathematics
Transactions 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
• Mathematics
Probability 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 $$C_1<d< c n/\log ^2 On the second eigenvalue of random bipartite biregular graphs We consider the spectral gap of a uniformly chosen random (d_1,d_2)-biregular bipartite graph G with |V_1|=n, |V_2|=m, where d_1,d_2 could possibly grow with n and m. Let A be the Sharp transition of the invertibility of the adjacency matrices of sparse random graphs • Mathematics • 2018 We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we The rank of random regular digraphs of constant degree • Mathematics, Computer Science J. Complex. • 2018 It is shown that A_n has rank at least at least n-1 with probability going to one as n goes to infinity. The circular law for sparse non-Hermitian matrices • Mathematics The Annals of Probability • 2019 For a class of sparse random matrices of the form A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n, where \{\xi_{i,j}\} are i.i.d.~centered sub-Gaussian random variables of unit variance, and Singularity of sparse Bernoulli matrices • Mathematics • 2020 Let M_n be an n\times n random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant C\geq 1 such that, whenever p and n satisfy C\log n/n\leq p\leq C^{-1}, Singularity of Bernoulli matrices in the sparse regime pn = O(\log(n)) Consider an n\times n random matrix A_n with i.i.d Bernoulli(p) entries. In a recent result of Litvak-Tikhomirov, they proved the conjecture$$ \mathbb{P}\{\mbox{$A_n$ is singular}\}=(1+o_n(1))
The sparse circular law under minimal assumptions
• Mathematics
Geometric 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
• Mathematics
Transactions 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
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 $$\min (d,n-d)\ge C\log The Circular Law for random regular digraphs • Nicholas A. Cook • Mathematics Annales de l'Institut Henri Poincaré, Probabilités et Statistiques • 2019 Let \log^Cn\le d\le n/2 for a sufficiently large constant C>0 and let A_n denote the adjacency matrix of a uniform random d-regular directed graph on n vertices. We prove that as n tends Adjacency matrices of random digraphs: singularity and anti-concentration • Mathematics • 2015 Let {\mathcal D}_{n,d} be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen uniformly at random from {\mathcal D}_{n,d} and M be its adjacency matrix. We The smallest singular value of a shifted d-regular random square matrix • Mathematics Probability Theory and Related Fields • 2018 We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let$$C_1<d< c n/\log ^2
The spectral gap of dense random regular graphs
• Mathematics
The 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, Physics Communications 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 • Mathematics The 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