# 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
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 randomExpand
Invertibility of adjacency matrices for random d-regular graphs
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 aExpand
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 weExpand
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 $$\mathbb{P}\{\mbox{A_n is singular}\}=(1+o_n(1))Expand The sparse circular law under minimal assumptions • Mathematics • Geometric and Functional Analysis • 2019 The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized$${n \times n}$$n×n matrix with i.i.d. entries converges to the uniform measure on the unit disc asExpand The Circular Law for random regular digraphs • Nicholas A. Cook • Mathematics • Annales de l'Institut Henri Poincaré, Probabilités et Statistiques • 2019 Let \log^Cn\le d\le n/2 for a sufficiently large constant C>0 and let A_n denote the adjacency matrix of a uniform random d-regular directed graph on n vertices. We prove that as n tendsExpand #### References SHOWING 1-10 OF 67 REFERENCES Adjacency matrices of random digraphs: singularity and anti-concentration • Mathematics • 2015 Let {\mathcal D}_{n,d} be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen uniformly at random from {\mathcal D}_{n,d} and M be its adjacency matrix. WeExpand Circular law for sparse random regular digraphs • Mathematics • 2018 Fix a constant C\geq 1 and let d=d(n) satisfy d\leq \ln^{C} n for every large integer n. Denote by A_n the adjacency matrix of a uniform random directed d-regular graph on n vertices.Expand On the singularity of adjacency matrices for random regular digraphs We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming$$\min (d,n-d)\ge C\logExpand
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 aExpand
Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition
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 ofExpand
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 areExpand
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 $\log^{12}n/(\log \log n)^{4} \leExpand Invertibility of Sparse non-Hermitian matrices • Mathematics • 2015 We consider a class of sparse random matrices of the form$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}\}$areExpand Sample covariance matrices of heavy-tailed distributions Let$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\leqExpand