## Duke Mathematical Journal

### Random symmetric matrices are almost surely nonsingular

#### Abstract

Let $Q_n$ denote a random symmetric ($n{\times}n)$-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values $0$ and $1$ with probability $1/2$). We prove that $Q_n$ is nonsingular with probability $1-O(n^{-1/8+\delta})$ for any fixed $\delta > 0$. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices

#### Article information

Source
Duke Math. J. Volume 135, Number 2 (2006), 395-413.

Dates
First available in Project Euclid: 17 October 2006

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1161093270

Digital Object Identifier
doi:10.1215/S0012-7094-06-13527-5

Mathematical Reviews number (MathSciNet)
MR2267289

Zentralblatt MATH identifier
1110.15020

Subjects
Primary: 15A52
Secondary: 05D40: Probabilistic methods

#### Citation

Costello, Kevin P.; Tao, Terence; Vu, Van. Random symmetric matrices are almost surely nonsingular. Duke Math. J. 135 (2006), no. 2, 395--413. doi:10.1215/S0012-7094-06-13527-5. http://projecteuclid.org/euclid.dmj/1161093270.

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