of $x$ such that $f(x)\underset{x\rightarrow-\infty}{\longrightarrow}0$ and $f(x)\underset{x\rightarrow+\infty}{\longrightarrow}1$, and if $f$ reaches 1 fast enough, namely $1-f(x)\underset{x\rightarrow+\infty}{\sim}e^{-\gamma x}$ with $\gamma>1$, then the traveling wave solution holds and the front position is still given by Eq. (\[eq:FKPPpos\]). Another interesting particular case is the “critical case” when $f$ is such that $\gamma=1$ exactly. Then, $$\left[\text{FKPP front position}\right]_\text{critical}=2t-\frac{1}{2}\ln t
+\text{const}
+{\frac{C_Y}{\sqrt{t}}}+\cdots
\label{eq:FKPPposcrit}$$ where $C_Y$ is a constant that we shall determine later on (see Sec. \[sec:det\]).\
There also exists a theorem established by Lalley and Sellke [@LS:1987] that gives the asymptotic (large time) shape of the distribution of the position of the rightmost particle in a frame whose origin is at position $\ln Z$, where $Z$ is some random variable that depends on the realization and may be thought of as a characterization of the position of the bulk of the particles in the BRW. (Its precise definition will be given later on). More recently [@BD:2009; @BD:2011; @Aidekon:2013], the distribution of the distances between the foremost particles was derived with the help of the solution to the FKPP equation with some peculiar initial condition.\
In this paper, we propose a phenomenological picture of the fluctuations in BRW, and we derive within this picture some 