  We investigate the probabilities of large deviations for the position of the
front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice
in which $Y$ particles do not move while $X$ particles move as independent
simple continuous time random walks of total jump rate $2$. For a wide class of
initial conditions, we prove that a large deviations principle holds and we
show that the zero set of the rate function is the interval $[0,v]$, where $v$
is the velocity of the front given by the law of large numbers. We also give
more precise estimates for the rate of decay of the slowdown probabilities. Our
results indicate a gapless property of the generator of the process as seen
from the front, as it happens in the context of nonlinear diffusion equations
describing the propagation of a pulled front into an unstable state.
