new statistical properties of a random variable similar to $\ln Z$. (Appendix \[sec:Z\] also lists some properties of $\ln Z$ itself.)\
In Sec. \[sec:model\], we shall introduce our phenomenological picture for branching random walks. Section \[sec:distrib\] is devoted to deriving the quantitative predictions of this model for a particular random variable that can characterize the early-time fluctuations of the branching random walk. The computation of a few free constant parameters requires us to solve deterministic equations: This is explained in Sec. \[sec:det\]. Numerical checks are in order since our analytical results are based on conjectures: This is done in Sec. \[sec:numerics\]. In light of our phenomenological model, we shall then come back to the discussion of the Ebert-Van Saarloos result on the $1/\sqrt{t}$ correction to the position of FKPP fronts (Sec. \[sec:EVS\]). Conclusions are given in Sec. \[sec:conclusion\].

\[sec:model\] Phenomenological description of branching random walks
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Picture
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The picture of the fluctuations in branching random walks (BRWs) that we have in mind is the following. There are essentially three types of fluctuations that may affect the position of the front or of the foremost particle.

1.  First, there are fluctuations occurring at very early times ($t\sim 1$), when the 