article published yes Zach Hunter author Matthew Alan Kwan author 5fca0887-a1db-11eb-95d1-ca9d5e0453b30000-0002-4003-7567 Lisa Sauermann author Mehtaab Sawhney author MaKw department Randomness and structure in combinatorics project 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). https://research-explorer.ista.ac.at/download/22147/22151/2026_IMRN_Hunter.pdf application/pdfno Oxford University Press2026 eng 1073-7928 1687-0247 2510.1293310.1093/imrn/rnag126 202612 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>. Z. Hunter, M.A. Kwan, L. Sauermann, M. Sawhney, International Mathematics Research Notices 2026 (2026). Hunter Z, Kwan MA, Sauermann L, Sawhney M. 2026. On random matrices with large corank. International Mathematics Research Notices. 2026(12), rnag126. 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> 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> 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. 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>. 221472026-06-28T22:01:35Z2026-06-29T09:19:14Z