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Wednesday, January 27, 2016 - 15:30 to 16:30
Professor Rahul Roy
Indian Statistical Institute, New Delhi
Rank of random matrices
Abstract: Consider an $n\times n$ matrix, $G_n$, each of its entries is independently $0$ or $1$ with probability $1/2$ each. Koml\'{o}s (1967) showed that the probability that $G_n$ is nonsingular goes to $1$ as $n \to \infty$. The asymptotics can be better understood now and we show that $P(G_n \mbox{ is singular}) = O(n^{-1/2})$ as $n \to \infty$. In case we consider the symmetric $n\times n$ matrix, $W_n$, where each of its upper-triangular entries is independently $0$ or $1$ with probability $1/2$ each, then Costello, Tao and Vu (2006) showed that $P(W_n \mbox{ is singular}) = O(n^{-1/8})$ as $n \to \infty$ and Manrique, P\'{e}rez-Abreu and Roy (2016) improved this result to obtain $P(W_n \mbox{ is singular}) = O(n^{-1/4 + \epsilon})$ for any $\epsilon > 0$ as $n \to \infty$. The proofs of these results depend primarily on a combinatorial identity and on a concentration inequality in probability. We discuss these in the two lectures.

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