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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the singularity probability of random Bernoulli matrices
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by Terence Tao and Van Vu PDF
J. Amer. Math. Soc. 20 (2007), 603-628 Request permission

Abstract:

Let $n$ be a large integer and let $M_n$ be a random $n \times n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability $1/2$). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Komlós and Szemerédi, as well as earlier work by the authors. The key new ingredient is the applications of Freiman-type inverse theorems and other tools from additive combinatorics.
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Additional Information
  • Terence Tao
  • Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
  • MR Author ID: 361755
  • ORCID: 0000-0002-0140-7641
  • Email: tao@math.ucla.edu
  • Van Vu
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
  • Email: vanvu@ucsd.edu
  • Received by editor(s): November 5, 2004
  • Published electronically: February 6, 2007
  • Additional Notes: The first author is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.
    The second author is an A. Sloan Fellow and is supported by an NSF Career Grant.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 20 (2007), 603-628
  • MSC (2000): Primary 15A52
  • DOI: https://doi.org/10.1090/S0894-0347-07-00555-3
  • MathSciNet review: 2291914