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:
upon contact with an $X$ particle, a $Y$ particle
instantaneously becomes $X$. This model can be
interpreted as an infection process, where $X$ and
$Y$ particles represent ill and healthy individuals
respectively, or as a combustion reaction, where the
$X$ and $Y$ particles correspond to heat units and
reactive molecules respectively, modeling the
combustion of a propellant into a stable stationary
state. For a wide class of initial conditions we
prove that a large deviation principle holds for the
position of the front, defined as the rightmost site
visited by an $X$ particle at a given time, showing
that the zero set of the rate function is the
interval $[0,v]$, where $v$ is the velocity given
by the law of large numbers. We also give more
precise estimates for the rate of decay of the
slowdown probabilities. Our results are in
agreement with the phenomenon of slow < (algebraic)
relaxation of the velocity for pulled nonlinear
diffusion equations studied in the physics
literature, indicating a gapless property of the
generator of the process. This is a joint work with
Jean B\'erard.
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