How Frequently is a System of 2-Linear Boolean Equations Solvable?

We consider a random system of equations xi + xj = b(i,j)(mod 2), (xu ∈ {0, 1}, b(u,v) = b(v,u) ∈ {0, 1}), with the pairs (i, j) from E, a symmetric subset of [n]× [n]. E is chosen uniformly at random among all such subsets of a given cardinality m; alternatively (i, j) ∈ E with a given probability p, independently of all other pairs. Also, given E, Pr{be = 0} = Pr{be = 1} for each e ∈ E, independently of all other be′ . It is well known that, as m passes through n/2 (p passes through 1/n, resp.), the underlying random graph G(n,#edges = m), (G(n, Pr(edge) = p), resp.) undergoes a rapid transition, from essentially a forest of many small trees to a graph with one large, multicyclic, component in a sea of small tree components. We should expect then that the solvability probability decreases precipitously in the vicinity of m ∼ n/2 (p ∼ 1/n), and indeed this probability is of order (1−2m/n)1/4, for m < n/2 ((1− pn)1/4, for p < 1/n, resp.). We show that in a near-critical phase m = (n/2)(1+λn−1/3) (p = (1+λn−1/3)/n, resp.), λ = o(n1/12), the system is solvable with probability asymptotic to c(λ)n−1/12, for some explicit function c(λ) > 0. Mike Molloy noticed that the Boolean system with be ≡ 1 is solvable iff the underlying graph is 2-colorable, and asked whether this connection might be used to determine an order of probability of 2-colorability in the near-critical case. We answer Molloy’s question affirmatively and show that, for λ = o(n1/12), the probability of 2-colorability is . 2−1/4e1/8c(λ)n−1/12, and asymptotic to 2−1/4e1/8c(λ)n−1/12 at a critical phase λ = O(1), and for λ → −∞.