Finding cores of random 2-SAT formulae via Poisson cloning

For the random 2-SAT formula F (n, p), let FC(n, p) be the formula left after the pure literal algorithm applied to F (n, p) stops. Using the recently developed Poisson cloning model together with the cut-off line algorithm (COLA), we completely analyze the structure of FC(n, p). In particular, it is shown that, for λ := p(2n− 1) = 1 + σ with σ ≫ n−1/3, the core of F (n, p) has θ2 λ n+O((θ λ n)1/2) variables and θ2 λ λn+O((θ λ n))1/2 clauses, with high probability, where θ λ is the larger solution of the equation θ − (1 − eλ) = 0. We also estimate the probability of F (n, p) being satisfiable to obtain Pr[F2(n, λ 2n−1) is satisfiable] = 