The discrepancy of random rectangular matrices

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