The determinant magnitude conjecture for random sign matrices

Let MnM_n be an n×nn\times n random matrix with independent Rademacher entries. Hadamard's inequality gives detMnnn/2|\det M_n|\leq n^{n/2}.

Determinant magnitude conjecture. Almost surely,

detMn=n(1/2o(1))n.|\det M_n|=n^{(1/2-o(1))n}.

The source states that Tao and Vu established a matching lower bound, confirming this conjecture, so it is solved.

Sources & referencesView supporting material

Primary source

Van Vu, “Recent progress in combinatorial random matrix theory”, arXiv:2005.02797 (2020).

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