## Random symmetric matrices are almost surely nonsingular.(English)Zbl 1110.15020

The authors consider random symmetric $$n \times n$$ matrices $$Q_n$$ whose upper-diagonal entries are independent, identically distributed, and take values $$0$$ and $$1$$ with probabilities $$1/2$$. Asymptotical relations are obtained presenting upper estimates of the probability $$p_n$$ that $$Q_n$$ is non-singular.
Theorem 1 states that for any $$\delta > 0$$ for sufficiently large $$n$$ the probability $$p_n \leq {an}^{-1/8 +\delta},$$ where $$a = a (\delta)$$ does not depend on $$n$$.
This assertion is generalized as follows. A random variable $$\xi$$ is said to have $$\rho$$-property if $\max_ {c \in \mathbb R}{\mathbf P}(\xi = c ) \leq \rho.$ Theorem 2 states the following. Let some $$\rho > 0$$ exist such that in the sequence of matrices $$Q_n = \{\xi_{ij}\}$$ all entries $$\xi_{ij}$$ have $$\rho$$-property. Then for any $$\delta > 0$$ for sufficiently large $$n$$, we have $$p_{n} \leq bn^{-1/8 + \delta},$$ where $$b = b (\rho, \delta)$$ is independent of $$n$$.
One more theorem presents a quadratic generalization of the Littlewood-Offord inequality. Let $$z_1,z_2, \dots, z_n$$ be independent identically distributed random variables that are equal to $$0$$ or $$1$$ with the probabilities $$1/2$$. Consider quadratic forms $Q = \sum^{n}_{i,j=1} c_{ij}z_i z_j,$ in which sufficiently many coeficients are larger $$1$$ in absolute value. The authors obtain an asymptotical upper estimate for the probability $${\mathbf P}(Q \in I)$$ as $$n \rightarrow \infty$$, where $$I$$ is an arbitrary non-random interval of length $$1$$.

### MSC:

 15B52 Random matrices (algebraic aspects) 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 60E15 Inequalities; stochastic orderings

### Keywords:

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