The discrepancy of random rectangular matrices

Dylan J. Altschuler, Jonathan Niles-Weed

Research output: Contribution to journalArticlepeer-review


A recent approach to the Beck–Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoß (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exchangeable pairs. While previous approaches are limited to dense matrices, our techniques allow us to work with matrices of all densities. This may be of independent interest for other sparse random constraint satisfaction problems.

Original languageEnglish (US)
Pages (from-to)551-593
Number of pages43
JournalRandom Structures and Algorithms
Issue number4
StatePublished - Jul 2022


  • Stein's method
  • discrepancy
  • random constraint satisfaction
  • sparse random graphs

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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