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Singularity of random Bernoulli matrices
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abstract
For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$ is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some generalizations are considered.
Forward citations
Cited by 2 Pith papers
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An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
If a quadratic Bernoulli polynomial has a point probability much larger than 1/n, it is close to a quadratic form of low rank; a consequence bounds edge-count point probabilities in Ramsey graphs by n^{-1+o(1)}.
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Quantitative invertibility of random matrices: a combinatorial perspective
For any fixed n×n complex matrix of norm up to 2^{n^0.001}, the least singular value of the matrix plus i.i.d. centered unit-variance complex noise is smaller than η with probability at most C(ξ) α, for α as small as ...
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