---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22147'
abstract:
- lang: eng
  text: "Let 1 ≤ k ≤ n and M be a random n × n matrix with independent uniformly random
    {±1}-entries. We\r\nshow that there exists an absolute constant c > 0 such that\r\nP[rank(M)
    ≤ n − k] ≤ exp(−cnk).\r\nThis confirms a well-known prediction in the area, extending
    a result of Rudelson (who previously\r\nproved this same result under the restriction
    k ≤ √n, via different methods)."
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\r\nResearch Foundation)—CRC 1720–539309657.
  This research was conducted during the period M.S. served\r\nas a Clay Research
  Fellow. This work began when the authors were visiting Mathematisches Forschungsinstitut
  Oberwolfach, which\r\nprovided ideal working conditions. M.S. thanks Vishesh Jain
  for initial discussions regarding the problem.\r\nWe also thank the anonymous referee
  for helpful comments."
article_number: rnag126
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Zach
  full_name: Hunter, Zach
  last_name: Hunter
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
- first_name: Lisa
  full_name: Sauermann, Lisa
  last_name: Sauermann
- first_name: Mehtaab
  full_name: Sawhney, Mehtaab
  last_name: Sawhney
citation:
  ama: Hunter Z, Kwan MA, Sauermann L, Sawhney M. On random matrices with large corank.
    <i>International Mathematics Research Notices</i>. 2026;2026(12). doi:<a href="https://doi.org/10.1093/imrn/rnag126">10.1093/imrn/rnag126</a>
  apa: Hunter, Z., Kwan, M. A., Sauermann, L., &#38; Sawhney, M. (2026). On random
    matrices with large corank. <i>International Mathematics Research Notices</i>.
    Oxford University Press. <a href="https://doi.org/10.1093/imrn/rnag126">https://doi.org/10.1093/imrn/rnag126</a>
  chicago: Hunter, Zach, Matthew Alan Kwan, Lisa Sauermann, and Mehtaab Sawhney. “On
    Random Matrices with Large Corank.” <i>International Mathematics Research Notices</i>.
    Oxford University Press, 2026. <a href="https://doi.org/10.1093/imrn/rnag126">https://doi.org/10.1093/imrn/rnag126</a>.
  ieee: Z. Hunter, M. A. Kwan, L. Sauermann, and M. Sawhney, “On random matrices with
    large corank,” <i>International Mathematics Research Notices</i>, vol. 2026, no.
    12. Oxford University Press, 2026.
  ista: Hunter Z, Kwan MA, Sauermann L, Sawhney M. 2026. On random matrices with large
    corank. International Mathematics Research Notices. 2026(12), rnag126.
  mla: Hunter, Zach, et al. “On Random Matrices with Large Corank.” <i>International
    Mathematics Research Notices</i>, vol. 2026, no. 12, rnag126, Oxford University
    Press, 2026, doi:<a href="https://doi.org/10.1093/imrn/rnag126">10.1093/imrn/rnag126</a>.
  short: Z. Hunter, M.A. Kwan, L. Sauermann, M. Sawhney, International Mathematics
    Research Notices 2026 (2026).
corr_author: '1'
das_tickbox: '0'
date_created: 2026-06-28T22:01:35Z
date_published: 2026-06-01T00:00:00Z
date_updated: 2026-06-29T09:19:14Z
day: '01'
ddc:
- '500'
department:
- _id: MaKw
doi: 10.1093/imrn/rnag126
external_id:
  arxiv:
  - '2510.12933'
file:
- access_level: open_access
  checksum: 396b47d0532d7ea509f8cd30f11392a8
  content_type: application/pdf
  creator: dernst
  date_created: 2026-06-29T09:15:15Z
  date_updated: 2026-06-29T09:15:15Z
  file_id: '22151'
  file_name: 2026_IMRN_Hunter.pdf
  file_size: 524993
  relation: main_file
  success: 1
file_date_updated: 2026-06-29T09:15:15Z
has_accepted_license: '1'
intvolume: '      2026'
issue: '12'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: bd95085b-d553-11ed-ba76-e55d3349be45
  grant_number: '101076777'
  name: Randomness and structure in combinatorics
publication: International Mathematics Research Notices
publication_identifier:
  eissn:
  - 1687-0247
  issn:
  - 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: On random matrices with large corank
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2026
year: '2026'
...