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We have performed direct Monte Carlo simulations of the reversible diffusion-limited process A + A~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

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