# Structure of eigenvectors of random regular digraphs

@article{Litvak2019StructureOE, title={Structure of eigenvectors of random regular digraphs}, author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef}, journal={Transactions of the American Mathematical Society}, year={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 random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and…

## 13 Citations

Circular law for sparse random regular digraphs

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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.…

Invertibility of adjacency matrices for random d-regular graphs

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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*}…

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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…

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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…

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There is a universal constant C\geq 1 such that, whenever $p and $n$ satisfy C\log n/n/n\leq p-1, there is a singular value of $M_n$ such that it contains a zero row or column.

Cokernels of adjacency matrices of random $r$-regular graphs

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- 2018

We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of M\'esz\'aros. We…

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This paper setted the conjecture that an A_n is singular matrix with i.i.d Bernoulli entries satisfies the sparse regime when p satisfies 1-o_n(1) for some large constant $C>1$.

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Lower bounds on delocalization of null vectors and eigenvectors of random matrices with i.i.d real subgaussian entries of zero mean and unit variance are found.

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The eigenvectors and eigenvalues of random matrices with iid entries are studied and an optimal estimate of the probability that an iid matrix has simple spectrum is provided, improving a recent result of Ge.

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