Electronic Journal of Probability

Rank deficiency in sparse random GF$[2]$ matrices

Richard Darling, Mathew Penrose, Andrew Wade, and Sandy Zabell

Full-text: Open access


Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $\mathcal{N}(n,m)$ denote the number of left null vectors in $\{0,1\}^m$ for $M$ (including the zero vector), where addition is mod 2. We take $n, m \to \infty$, with $m/n \to \alpha > 0$, while the weight distribution converges weakly to that of a random variable $W$ on $\{3, 4, 5, \ldots\}$. Identifying $M$ with a hypergraph on $n$ vertices, we define the 2-core of $M$ as the terminal state of an iterative algorithm that deletes every row incident to a column of degree 1.<br /><br />We identify two thresholds $\alpha^*$ and $\underline{\alpha\mkern-4mu}\mkern4mu$, and describe them analytically in terms of the distribution of $W$. Threshold $\alpha^*$ marks the infimum of values of $\alpha$ at which $n^{-1} \log{\mathbb{E}[\mathcal{N} (n,m)}]$ converges to a positive limit, while $\underline{\alpha\mkern-4mu}\mkern4mu$ marks the infimum of values of $\alpha$ at which there is a 2-core of non-negligible size compared to $n$ having more rows than non-empty columns. We have $1/2 \leq \alpha^* \leq \underline{\alpha\mkern-4mu}\mkern4mu \leq 1$, and typically these inequalities are strict; for example when $W = 3$ almost surely, $\alpha^* \approx 0.8895$ and $\underline{\alpha\mkern-4mu}\mkern4mu \approx 0.9179$. The threshold of values of $\alpha$ for which $\mathcal{N}(n,m) \geq 2$ in probability lies in $[\alpha^*,\underline{\alpha\mkern-4mu}\mkern4mu]$ and is conjectured to equal $\underline{\alpha\mkern-4mu}\mkern4mu$. The random row-weight setting gives rise to interesting new phenomena not present in the case of non-random $W$ that has been the focus of previous work.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 83, 36 pp.

Accepted: 14 September 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C65: Hypergraphs 05C80: Random graphs [See also 60B20] 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F10: Large deviations

Random sparse matrix null vector hypercycle random allocation XORSAT phase transition hypergraph core random equations large deviations Ehrenfest model

This work is licensed under a Creative Commons Attribution 3.0 License.


Darling, Richard; Penrose, Mathew; Wade, Andrew; Zabell, Sandy. Rank deficiency in sparse random GF$[2]$ matrices. Electron. J. Probab. 19 (2014), paper no. 83, 36 pp. doi:10.1214/EJP.v19-2458. https://projecteuclid.org/euclid.ejp/1465065725

Export citation


  • Alamino, R. C.; Saad, D. Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach. Phys. Rev. E (3) 77 (2008), no. 6, 061123, 12 pp.
  • Balakin, G. V.; Kolchin, V. F.; Khokhlov, V. I. Hypercycles in a random hypergraph. (Russian) Diskret. Mat. 3 (1991), no. 3, 102–108; translation in Discrete Math. Appl. 2 (1992), no. 5, 563–570
  • Calkin, N. J. Dependent sets of constant weight binary vectors. Combin. Probab. Comput. 6 (1997), no. 3, 263–271.
  • Cooper, C. Asymptotics for dependent sums of random vectors. Random Structures Algorithms 14 (1999), no. 3, 267–292.
  • Cooper, C. The cores of random hypergraphs with a given degree sequence. Random Structures Algorithms 25 (2004), no. 4, 353–375.
  • Costello, K. P.; Vu, V. On the rank of random sparse matrices. Combin. Probab. Comput. 19 (2010), no. 3, 321–342.
  • Darling, R. W. R.; Levin, D. A.; Norris, J. R. Continuous and discontinuous phase transitions in hypergraph processes. Random Structures Algorithms 24 (2004), no. 4, 397–419.
  • Darling, R. W. R.; Norris, J. R. Structure of large random hypergraphs. Ann. Appl. Probab. 15 (2005), no. 1A, 125–152.
  • Darling, R. W. R.; Norris, J. R. Differential equation approximations for Markov chains. Probab. Surv. 5 (2008), 37–79.
  • Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M.: Tight thresholds for cuckoo hashing via XORSAT. Proc. 37th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, 2010, Volume 6198/2010, Springer, pp. 213–225.
  • Dubois, O.; Mandler, J. The 3-XORSAT threshold. C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 963–966.
  • Feller, W. An Introduction to Probability Theory and Its Applications. Vol. I. Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
  • Ibrahimi, M., Kanoria, Y., Kraning, M. and Montanari, A.: The set of solutions of random XORSAT formulae. SODA `12 Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, 2012, pp. 760–779.
  • Kolchin, V. F.: Cycles in random graphs and hypergraphs (abstract). Adv. in Appl. Probab. 24, (1992), 768.
  • Kolchin, V. F. Random graphs and systems of linear equations in finite fields. Proceedings of the Fifth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science (PoznaÅ„, 1991). Random Structures Algorithms 5 (1994), no. 1, 135–146.
  • Kolchin, V. F. Random Graphs. Cambridge University Press, Cambridge, 1999. xii+252 pp. ISBN: 0-521-44081-5
  • Kolchin, V. F.; Sevast'yanov, B. A.; Chistyakov, V. P. Random Allocations. Translated from the Russian. Translation edited by A. V. Balakrishnan. V. H. Winston & Sons, Washington, D.C. 1978. xi+262 pp. ISBN: 0-470-99394-4
  • Kovalenko, I. N.; Levitskaya, A. A. Stochastic properties of systems of random linear equations over finite algebraic structures. Probabilistic methods in discrete mathematics (Petrozavodsk, 1992), 64–70, Progr. Pure Appl. Discrete Math., 1, VSP, Utrecht, 1993.
  • Levitskaya, A. A. Systems of random equations over finite algebraic structures. (Russian) Kibernet. Sistem. Anal. 41 (2005), no. 1, 82–116, 190; translation in Cybernet. Systems Anal. 41 (2005), no. 1, 67–93
  • Mahmoud, H. M. Pólya Urn Models. CRC Press, Boca Raton, FL, 2009. xii+290 pp. ISBN: 978-1-4200-5983-0.
  • Mézard, M., Parisi, G. and Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297 (2002), 812–815.
  • Muetze, T. Generalized switch-setting problems. Discrete Math. 307 (2007), no. 22, 2755–2770.
  • Talagrand, M. Spin Glasses: A Challenge for Mathematicians. Springer-Verlag, Berlin, 2003. x+586 pp. ISBN: 3-540-00356-8.