system consists in a few particles. They have a large (of order 1) and lasting impact on the position of the front or of the rightmost particle. The main effect is given by the random waiting time of the first particle before it splits into two particles, during which it diffuses, but the subsequent waiting times of the latter two particles also contribute, etc... until the system contains a large enough number of particles that makes it partly “deterministic”. We do not believe that there is a simple way to compute the effect of these fluctuations, since there is no large parameter in the problem which would allow for sensible approximations.

2.  Once the system contains many particles, which happens say at time $t_0^\prime\gg 1$, it enters a “mean-field” regime: In a first approximation, its particle density obeys a deterministic evolution with a moving absorptive boundary at a position that we shall call $\bar X_t$ (and symmetrically at $-\bar X_t$), set in such a way that the particle density is 1 at some fixed distance of order 1 to the left or to the right of this boundary, respectively. These boundaries simulate the discreteness of the particles. This is the 