The time constant vanishes only on the percolation cone in directed first passage percolation

We consider the directed first passage percolation model on Z2. In this model, we assign independently to each edge e a passage time t(e) with a common distribution F . We denote by ~ T (0, (r, θ)) the passage time from the origin to (r, θ) by a northeast path for (r, θ) ∈ R+ × [0, π/2]. It is known that ~ T (0, (r, θ))/r converges to a time constant ~μF (θ). Let ~ pc denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by ~ pc, as follows: (1) If F (0) < ~ pc, then ~μF (θ) > 0 for all 0 ≤ θ ≤ π/2. (2) If F (0) = ~ pc, then ~μF (θ) > 0 if and only if θ 6= π/4. (3) If F (0) = p > ~ pc, then there exists a percolation cone between θ − p and θ + p for 0 ≤ θ− p < θ+ p ≤ π/2 such that ~μ(θ) > 0 if and only if θ 6∈ [θ− p , θ+ p ]. Furthermore, all the moments of ~ T (0, (r, θ)) converge whenever θ ∈ [θ− p , θ+ p ]. As applications, we describe the shape of the directed growth model on the distribution of F . We give a phase transition for the shape divided by ~pc.