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

@article{Litvak2018TheSS, title={The smallest singular value of a shifted d-regular random square matrix}, author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef}, journal={Probability Theory and Related Fields}, year={2018}, volume={173}, pages={1301-1347} }

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 $$C_1<d< c n/\log ^2 n$$C1<d<cn/log2n and let $$\mathcal {M}_{n,d}$$Mn,d be the set of all $$n\times n$$n×n square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$Mn,d has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$Mn,d. Then… Expand

#### 18 Citations

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 $\kappa \geq 0$,… Expand

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.… Expand

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… Expand

Invertibility of adjacency matrices for random d-regular graphs

- Mathematics
- Duke 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*}… Expand

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… Expand

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 $C\log n/n\leq p\leq C^{-1}$,… Expand

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… Expand

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))… Expand

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… Expand

The Circular Law for random regular digraphs

- 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… Expand

#### References

SHOWING 1-10 OF 67 REFERENCES

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… Expand

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.… Expand

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… Expand

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… Expand

Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition

- Mathematics
- 2015

We obtain non-asymptotic lower bounds on the least singular value of ${\mathbf X}_{pn}^\top/\sqrt{n}$, where ${\mathbf X}_{pn}$ is a $p\times n$ random matrix whose columns are independent copies of… Expand

Bounding the smallest singular value of a random matrix without concentration

- Mathematics
- 2013

Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are… Expand

On the interval of fluctuation of the singular values of random matrices

- Computer Science, Mathematics
- ArXiv
- 2015

It is proved that with high probability A/A has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. Expand

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… Expand

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… Expand

Sample covariance matrices of heavy-tailed distributions

- Mathematics
- 2016

Let $p>2$, $B\geq 1$, $N\geq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\sup\limits_{a\in S^{n-1}}{\mathrm E}|\langle X,a\rangle|^p\leq… Expand