Fluctuations of the front in a stochastic combustion model

We consider an interacting particle system on the one dimensional lattice $\mathbf{Z}$ modeling combustion. The process depends on two integer parameters $2\le a<M<\mathrm{\infty }$. Particles move independently as continuous time simple symmetric random walks except that 1. When a particle jumps to a site which has not been previously visited by any particle, it branches into $a$ particles; 2. When a particle jumps to a site with $M$ particles, it is annihilated. We start from a configuration where all sites to the left of the origin have been previously visited and study the law of large numbers and central limit theorem for $r_t$, the rightmost visited site at time $t$. The proofs are based on the construction of a renewal structure leading to a definition of regeneration times for which good tail estimates can be performed.