Fluctuations of the front in a one dimensional model of $X+Y\to 2X$

We consider a 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 continuous time, simple symmetric random walks. $Y$ particles are transformed instantaneously to $X$ particles upon contact. We start with a fixed number $a\ge 1$ of $Y$ particles at each site to the right of the origin. We prove a central limit theorem for the rightmost visited site of the $X$ particles up to time $t$ and show that the law of the environment as seen from the front converges to a unique invariant measure.