distribution since they do not have time to develop their own front at time $t$.

This picture is parallel to the phenomenological model for front fluctuations proposed in Ref. [@Brunet:2005bz] in the context of the stochastic FKPP problem.

\[sec:var\]Variables
--------------------

To arrive at quantitative predictions for the behavior of the BRW, we need to introduce random variables that characterize the realizations. We shall discuss the following ones:

-   $X_t$, the position of the rightmost particle,

-   $Y_t=\sqrt{t}\sum_i e^{x_i(t)-2t}$, where the sum goes over all the particles in the system,

-   $Z_t=\sum_i \left[2t-x_i(t)\right]e^{x_i(t)-2t}$.

Throughout, we shall denote by $\langle A \rangle$ the statistical average (over realizations) of a given variable $A$ in the full stochastic model, and by $\bar{A}$ the value of this variable in a mean-field approximation of the same model with a discreteness cutoff at the tip. (These notations have already been used in Eq. (\[eq:meannBRW\]) and Eqs. (\[eq:shape\]), (\[eq:Xt\]) respectively.) Discrete sums over the particles will often be replaced by integrals wherever the particle density is large enough.\
Let us briefly comment on the random variables we have just introduced.

-   As already mentioned, $\langle X_t\rangle$ is related to the solution of the FKPP equation with the step 