Duke Mathematical Journal

Random symmetric matrices are almost surely nonsingular

Kevin P. Costello, Terence Tao, and Van Vu

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Let Qn denote a random symmetric (n×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 Qn is nonsingular with probability 1-O(n-1/8+δ) for any fixed δ>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

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

First available in Project Euclid: 17 October 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 05D40: Probabilistic methods


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. https://projecteuclid.org/euclid.dmj/1161093270

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