Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts

We propose a phenomenological description for the effect of a weak noise on the position of a front described by the Fisher-Kolmogorov-Petrovsky-Piscounov equation or any other traveling-wave equation in the same class. Our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front. Our parameter-free analytical predictions for the velocity of the front, its diffusion constant and higher cumulants of its position agree with numerical simulations.

Figure 1

Measured $L_{\mathrm{eff}}$, defined by Eq. (30), from which we have subtracted the width $L$ in the cutoff theory, as a function of $N$. The dotted line represents the leading terms $3\ln \ln N∕{\gamma }_0$; see Eq. (9). The subleading terms (31) of the solid line have been determined by a fit.

Figure 2

From top to bottom, the correction to the velocity given by the cutoff theory and the cumulants of orders 2–5 of the position of the front in the stochastic model. The numerical data are compared to our parameter-free analytical predictions (6), represented by the dotted line. The subleading terms of the solid lines are numerically the same as in Fig. 1; no further fit has been performed for the present figure.

Figure 3

The ratio of cumulants 2–5 divided by the correction to the velocity to the power $3∕2$. The dotted lines are the analytical prediction assuming only the cutoff theory (5b) for the velocity and the predictions (6) for the cumulants.