Random n x n matrices with independent entries bounded away from point masses have P(rank ≤ n-k) ≤ exp(-c n k) for some c>0.
A large deviation inequality for the rank of a random matrix.Ann
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For n x n i.i.d. Bernoulli(p) matrices, P(corank >= k) = (1-p + o_n(1))^{k n} when k = O(sqrt(log n)).
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Exponential Rank Bounds for Random Matrices
Random n x n matrices with independent entries bounded away from point masses have P(rank ≤ n-k) ≤ exp(-c n k) for some c>0.
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Rank deficiency of Bernoulli random matrices for growing corank
For n x n i.i.d. Bernoulli(p) matrices, P(corank >= k) = (1-p + o_n(1))^{k n} when k = O(sqrt(log n)).