Fluctuations and Stability of Fisher Waves

We have performed direct Monte Carlo simulations of the reversible diffusion-limited process $A+A\leftrightarrow A$ to study the effect of fluctuations on a propagating interface between stable and unstable phases. The mean-field description of this process, Fisher's reaction-diffusion equation, admits stable nonlinear wave fronts. We find that this mean-field description breaks down in spatial dimensions 1 and 2, while it appears to be qualitatively and quantitatively accurate at and above 4 dimensions. In particular, the interface width grows $\sim t^{1/2}$ in 1D (exact) and $\sim t^{0.272\pm 0.007}$ in 2D (numerical).