We study the effect of the noise due to microscopic fluctuations on the position of a one dimensional front propagating from a stable to an unstable region in the "linearly marginal stability case." By simulating a very simple system for which the effective number N of particles can be as large as N=10(150), we measure the N dependence of the diffusion constant D, of the front and the shift of its velocity N-upsilon. Our results indicate that D-N similar to (log N)(-3). They also confirm our recent claim that the shift of velocity scales like upsilon (min) - upsilon (N) similar or equal to K(log N)(-2) and indicate that the numerical value of K is very close to the analytical expression K-approx obtained in our previous work using a simple cut-off approximation. [References: 37]
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