On random matrices with large corank
Hunter Z, Kwan MA, Sauermann L, Sawhney M. 2026. On random matrices with large corank. International Mathematics Research Notices. 2026(12), rnag126.
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Author
Hunter, Zach;
Kwan, Matthew AlanISTA 
;
Sauermann, Lisa;
Sawhney, Mehtaab
;
Sauermann, Lisa;
Sawhney, MehtaabCorresponding author has ISTA affiliation
Department
Abstract
Let 1 ≤ k ≤ n and M be a random n × n matrix with independent uniformly random {±1}-entries. We
show that there exists an absolute constant c > 0 such that
P[rank(M) ≤ n − k] ≤ exp(−cnk).
This confirms a well-known prediction in the area, extending a result of Rudelson (who previously
proved this same result under the restriction k ≤ √n, via different methods).
Publishing Year
Date Published
2026-06-01
Journal Title
International Mathematics Research Notices
Publisher
Oxford University Press
Acknowledgement
Z.H. was supported by SNSF grant 200021-228014. M.K. was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. L.S. was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation)—CRC 1720–539309657. This research was conducted during the period M.S. served
as a Clay Research Fellow. This work began when the authors were visiting Mathematisches Forschungsinstitut Oberwolfach, which
provided ideal working conditions. M.S. thanks Vishesh Jain for initial discussions regarding the problem.
We also thank the anonymous referee for helpful comments.
Volume
2026
Issue
12
Article Number
rnag126
ISSN
eISSN
IST-REx-ID
Cite this
Hunter Z, Kwan MA, Sauermann L, Sawhney M. On random matrices with large corank. International Mathematics Research Notices. 2026;2026(12). doi:10.1093/imrn/rnag126
Hunter, Z., Kwan, M. A., Sauermann, L., & Sawhney, M. (2026). On random matrices with large corank. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnag126
Hunter, Zach, Matthew Alan Kwan, Lisa Sauermann, and Mehtaab Sawhney. “On Random Matrices with Large Corank.” International Mathematics Research Notices. Oxford University Press, 2026. https://doi.org/10.1093/imrn/rnag126.
Z. Hunter, M. A. Kwan, L. Sauermann, and M. Sawhney, “On random matrices with large corank,” International Mathematics Research Notices, vol. 2026, no. 12. Oxford University Press, 2026.
Hunter Z, Kwan MA, Sauermann L, Sawhney M. 2026. On random matrices with large corank. International Mathematics Research Notices. 2026(12), rnag126.
Hunter, Zach, et al. “On Random Matrices with Large Corank.” International Mathematics Research Notices, vol. 2026, no. 12, rnag126, Oxford University Press, 2026, doi:10.1093/imrn/rnag126.
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