# Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

@article{Basak2018SharpTO, title={Sharp transition of the invertibility of the adjacency matrices of sparse random graphs}, author={Anirban Basak and Mark Rudelson}, journal={Probability Theory and Related Fields}, year={2018}, volume={180}, pages={233 - 308} }

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies np≥logn+k(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$np\ge \log n+k(n)$$\end{document…

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## References

SHOWING 1-10 OF 65 REFERENCES

### Singularity of discrete random matrices II

- Mathematics
- 2020

Let $\xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are independent copies of $\xi$. We show that,…

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

- Mathematics, Computer Science
- 2020

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 $C>1$.

### Singularity of sparse Bernoulli matrices

- Mathematics, Computer Science
- 2020

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.

### 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…

### Circular law theorem for random Markov matrices

- Mathematics
- 2008

Let (Xjk)jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ2. Let M be the n × n random Markov matrix with i.i.d. rows defined by…

### Investigate Invertibility of Sparse Symmetric Matrix

- Mathematics
- 2017

In this paper, we investigate the invertibility of sparse symmetric matrices. We will show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \delta_{ij} \xi_{ij}$ is…

### The circular law for sparse non-Hermitian matrices

- Mathematics, Computer ScienceThe 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…

### On the singularity of adjacency matrices for random regular digraphs

- Mathematics, Computer Science
- 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…