demonstrate that this a priori approximation is indeed accurate enough.

Let us consider a fluctuation occurring at time $t_1\gg 1$ at a distance $\delta$ from the tip of the deterministic front. From Eq. (\[eq:Xt\]), at time $t>t_1$ such that $t-t_1\gg1$, this fluctuation has developed its own front whose tip sits at position $$\bar X_{\delta,t}=\bar X_{t_1}+\delta+\bar X_{t-t_1}
=\bar X_t+\delta-\frac32 \ln\frac{t_1(t-t_1)}{t}.
\label{eq:Xdelta}$$ There would of course be terms proportional to $1/\sqrt{t_1}$, $1/\sqrt{t}$ and $1/\sqrt{t-t_1}$ also here, but we again anticipate that they would eventually lead to corrections of higher order to the quantities of interest. We refer the reader to Appendix A for the details.

The shape of the front generated by this fluctuation will eventually have the form $C\times \psi_{\bar X_{\delta,t}}(x,t-t_1)$, where $C$ is a constant that we cannot determine since it is related to some “average” shape of the fluctuation. With this extra fluctuation, $Y_t$ has the following expression: Y\_t=\_[-]{}\^[+]{}dx \_[|X\_t]{}(x,t)e\^[x-2t]{} +C\_[-]{}\^[+]{}dx \_[|X\_[,t]{}]{}(x,t-t\_1)e\^[x-2t]{} . The first term is just $\bar Y_t$: We replace it by Eq. (\[eq:Y00\]). The second term is integrated in the same way as the first one, using the expression (\[eq:Xdelta\]) for $\bar X_{\delta,t}$. We find Y\_t=-2(1+C ). Thus the forward shift in $\ln Y_t$ induced at 