Rank deficiency in sparse random GF[2] matrices

Darling, Richard W.R.; Penrose, Mathew D.; Wade, Andrew R.; Zabell, Sandy L.

Abstract

Let M be a random m×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 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→∞, with m/n→α>0, while the weight distribution converges weakly to that of a random variable W on {3,4,5,…}. 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. We identify two thresholds α∗ and α−, and describe them analytically in terms of the distribution of W. Threshold α∗ marks the infimum of values of α at which n−1logE[N(n,m)] converges to a positive limit, while α− marks the infimum of values of α 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≤α∗≤α−≤1, and typically these inequalities are strict; for example when W=3 almost surely, α∗≈0.8895 and α−≈0.9179. The threshold of values of α for which N(n,m)≥2 in probability lies in [α∗,α−] and is conjectured to equal α−. 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.

Citation

Darling, R. W., Penrose, M. D., Wade, A. R., & Zabell, S. L. (2014). Rank deficiency in sparse random GF[2] matrices. Electronic Journal of Probability, 19, Article 83. https://doi.org/10.1214/ejp.v19-2458

• Journal Article Type: Article
• Acceptance Date: Sep 4, 2014
• Online Publication Date: Sep 14, 2014
• Publication Date: Sep 14, 2014
• Deposit Date: Oct 5, 2014
• Publicly Available Date: Oct 6, 2014
• Journal: Electronic Journal of Probability
• Electronic ISSN: 1083-6489
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Article Number: 83
• DOI: https://doi.org/10.1214/ejp.v19-2458
• Public URL: https://durham-repository.worktribe.com/output/1420067