Rank deficiency in sparse random GF[2] matrices

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 $N(n,m)$ 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 may vary with $n$ but converges weakly to a limiting distribution on ${3, 4, 5, ...}$; let $W$ denote a variable with this limiting distribution. 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 $\underline{\alpha}$, and describe them analytically in terms of the distribution of $W$. Threshold $\alpha^*$ marks the infimum of values of $\alpha$ at which $n^{-1} \log{\mathbb{E} [N(n,m)}]$ converges to a positive limit, while $\underline{\alpha}$ 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 $1/2 \leq \alpha^* \leq \underline{\alpha} \leq 1$, and typically these inequalities are strict; for example when $W = 3$ almost surely, numerics give $\alpha^* = 0.88949 ...$ and $\underline{\alpha} = 0.91793 ...$ (previous work on this model has mainly been concerned with such cases where $W$ is non-random). The threshold of values of $\alpha$ for which $N(n,m) \geq 2$ in probability lies in $[\alpha^*,\underline{\alpha}]$ and is conjectured to equal $\underline{\alpha}$. The random row weight setting gives rise to interesting new phenomena not present in the non-random case that has been the focus of previous work.