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.
Highlights
Suppose that M := M (n, m) is an m × n matrix with entries in {0, 1}, each of whose rows contains at least one 1, for which we seek a left null vector over GF[2], i.e. a row vector a ∈ {0, 1}m such that aM ≡ 0, where here and elsewhere 0 is the all-0 vector
We study asymptotics of the expected size E[N (n, m)] and probability P[N (n, m) > 1] of non-triviality of the left null space of M (n, m), in terms of the asymptotic aspect ratio α
We study the rate of exponential decay of the probability that 1 := (1, 1, . . . , 1) is a null vector
Summary
Suppose that M := M (n, m) is an m × n matrix with entries in {0, 1}, each of whose rows contains at least one 1, for which we seek a left null vector over GF[2], i.e. a row vector a ∈ {0, 1}m such that aM ≡ 0 (mod 2), where here and elsewhere 0 is the all-0 vector. The general case, allowing other weight distributions, corresponds to a generalization of the Ehrenfest model whereby multiple ‘diffusions’ are allowed, i.e. at each step several particles may change colour at once; cf [20, Chapter 10]. This can be interpreted in terms of a walk on a version of the hypercube with additional edges. Problems may be formulated in terms of random hypergraphs: each row represents a hyperedge, and each column represents a vertex (see Section 2.3 below).
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