On the Rank of a Random Binary Matrix

Abstract

We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \choose k}$ such columns.
 We obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn/k$. The matrix $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m<m^*$ the rank of $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$.
 In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$, the expected length of a minimum weight spanning tree tends to $\zeta(3)\sim 1.202$.

Abelian groups from random hypergraphs

“…For this same c d , as is pointed out in [11], Pittel and Sorkin show that a random n × m matrix over Z/2Z with exactly d + 1 1's in each column will have nontrivial kernel when the average number of 1's in each row exceeds c d and trivial kernel when this average is below c d . Work of Cooper, Frieze, and Pegden [6] refines this result to describe the asymptotic rank of the random Z/2Z matrix on either side of the phase transition. It is at the phase transition of [17] that experiments witness a torsion burst in our random matrix model.…”

“…For the proof of Theorem 4, we use Theorem 2 as well as a theorem of Cooper, Frieze, and Pegden [6]. In [6], the authors consider the same model that we consider except that the matrices are taken over rather than over .…”

“…Also because of the comparison between and we prove the following analogue to the homology vanishing threshold in our random matrix model. This proof will follow immediately from Theorem 2 and a result about the version of the random matrix model from [6].…”

“…Random matrices with a fixed number of nonzero entries in each row have been considered in the past and [11] in particular mentions their similarity with . For example Pittel and Sorkin [17] consider this same model for random matrices we describe here over in studying random instances of -XORSAT and Cooper, Frieze, and Pegden [6] prove that a random matrix over with exactly 1’s in each column exhibit a phase transition in the rank. The phase transitions described in [6, 17] are perfectly analogous to the phase transition in homology of established by [2, 11] in quite a strong sense.…”

“…For example Pittel and Sorkin [17] consider this same model for random matrices we describe here over in studying random instances of -XORSAT and Cooper, Frieze, and Pegden [6] prove that a random matrix over with exactly 1’s in each column exhibit a phase transition in the rank. The phase transitions described in [6, 17] are perfectly analogous to the phase transition in homology of established by [2, 11] in quite a strong sense. In with constant, if is large enough that the average degree of a -dimensional face exceeds then asymptotically almost surely has nonvanishing th homology while for small enough that the average degree of -dimensional face is smaller than then -homology is generated by -simplex boundaries.…”