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

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1.	Title	Title of document	Rank deficiency in sparse random GF$[2]$ matrices
2.	Creator	Author's name, affiliation, country	Richard W. R. Darling; National Security Agency; United States
2.	Creator	Author's name, affiliation, country	Mathew D. Penrose; University of Bath; United Kingdom
2.	Creator	Author's name, affiliation, country	Andrew R. Wade; University of Durham; United Kingdom
2.	Creator	Author's name, affiliation, country	Sandy L. Zabell; Northwestern University; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Random sparse matrix, null vector, hypercycle, random allocation, XORSAT, phase transition, hypergraph core, random equations, large deviations, Ehrenfest model
3.	Subject	Subject classification	60C05 (Primary) 05C65; 05C80; 15B52; 60B20; 60F10
4.	Description	Abstract	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. 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2014-09-14
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/2458
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-2458
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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