Distribution tail of the distance to infinity in supercritical bond percolation

We give an estimate for the distribution tail of the passage time between $0$ and $\mathrm{\infty }$ in first-passage percolation on ${\mathbb{Z}}^d$, when the weight distribution is bounded and has a supercritical atom at $0$. A special case involves the number of disjoint cutsets between $0$ and $\mathrm{\infty }$ in supercritical bond percolation. Besides, we show that the rate function for the upper-tail deviations of the point-to-point passage time is asymptotically polynomial near $0$.