function $\psi_{\bar X}$ itself, but it turns out that we do not need to take them into account in our model, except for the determination of one overall numerical constant: We will come back to the derivation of Eq. (\[eq:shape\]) and of its corrections in Sec. \[sec:det\].)

    The fluctuations on top of this essentially deterministic front we have just described must take place in the tip region, where the particle density is low. We shall assume that a single fluctuation effectively gives the dominant correction to the deterministic evolution, and that the distribution $p(\delta)$ of the position $\delta$ of this fluctuation with respect to the tip of the front is exponential: $$p(\delta)=C_1 e^{-\delta}.
    \label{eq:proba}$$ We have found (see below) that these fluctuations bring a contribution of order $1/\sqrt{t}$ to the average position both of the front and of the rightmost particle in the BRW.

3.  Finally, there are tip fluctuations occurring at very late times, say between $t-\bar t_0$ and $t$, where $\bar t_0$ is of order 1. They are also distributed as $e^{-\delta}$. They obviously add noise to the position of the tip of the front, but they do not have an effect on the bulk of the particle 