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Seminar
Submitted by nabinjana on 14 January, 2016 - 15:52
Date/Time:
Thursday, January 28, 2016 - 15:30 to 16:30
Venue:
M1
Speaker:
Professor Rahul Roy
Affiliation:
Indian Statistical Institute, New Delhi
Title:
Rank of random matrices (continuation of last seminar)
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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