  Consider a discrete-time one-dimensional supercritical branching random walk.
We study the probability that there exists an infinite ray in the branching
random walk that always lies above the line of slope $\gamma-\epsilon$, where
$\gamma$ denotes the asymptotic speed of the right-most position in the
branching random walk. Under mild general assumptions upon the distribution of
the branching random walk, we prove that when $\epsilon\to 0$, the probability
in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where
$\beta$ is a positive constant depending on the distribution of the branching
random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables
(with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers an open
question of Robin Pemantle.
