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The discrepancy of random rectangular matrices
Author(s) -
Altschuler Dylan J.,
NilesWeed Jonathan
Publication year - 2022
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21054
Subject(s) - poisson distribution , constraint satisfaction problem , mathematics , bernoulli's principle , conjecture , combinatorics , random matrix , constant (computer programming) , constraint (computer aided design) , discrete mathematics , computer science , statistics , geometry , eigenvalues and eigenvectors , physics , quantum mechanics , probabilistic logic , programming language , engineering , aerospace engineering
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.