realizations. Second, a theoretical drawback: It turns out that the finite-time corrections to the moments of $\ln Z_t$ are very sensitive to the initial fluctuations, the ones that are not computable analytically. We shall nevertheless quote a few results on the distribution of $\ln Z_t$ in Appendix B.

In some intuitive sense, $\ln Y_t$ and $\ln Z_t$ characterize the position of the “front” of a particular realization of the evolution at time $t$.\
The variables $\ln Y_t$ and $\ln Z_t$ keep the memory of the initial fluctuations. Therefore, we shall not attempt to compute the distribution of the fluctuations in $\ln Y$ accumulated over the whole history of the BRW, but instead the fluctuations of this variable between two large times $t_0$ and $t$, in order to have a quantity that is independent of the very early times at which there is no mean-field regime.

\[sec:distrib\]Statistics of $f\equiv \ln Y_t-\ln Y_{t_0}$ in the phenomenological picture 
===========================================================================================

Here, starting from the phenomenological model defined in Sec. \[sec:model\], we shall deduce new results on the distribution $p(f)$ of the variable $f\equiv \ln Y_t-\ln Y_{t_0}$ (and on its moments) for $t,t_0,t-t_0\gg 1$, up to one single constant for the moments of order larger 