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2014 Rank deficiency in sparse random GF$[2]$ matrices
Richard Darling, Mathew Penrose, Andrew Wade, Sandy Zabell
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Electron. J. Probab. 19: 1-36 (2014). DOI: 10.1214/EJP.v19-2458

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.<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.

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Richard Darling. Mathew Penrose. Andrew Wade. Sandy Zabell. "Rank deficiency in sparse random GF$[2]$ matrices." Electron. J. Probab. 19 1 - 36, 2014. https://doi.org/10.1214/EJP.v19-2458

Information

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

zbMATH: 1352.60010
MathSciNet: MR3263640
Digital Object Identifier: 10.1214/EJP.v19-2458

Subjects:
Primary: 60C05
Secondary: 05C65 , 05C80 , 15B52 , 60B20 , 60F10

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

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