this section, we first review the Ebert-Van Saarloos method [@EVS:2000] to compute the order $1/\sqrt{t}$ correction to the mean position of the rightmost particle in the BRW $\langle X_t\rangle$. We extend the method to the position of the right boundary in the deterministic model with discreteness cutoffs $\bar X_t$, and eventually adapt it to $\langle \ln Y_t\rangle$.

The calculations presented here will enable us to determine the remaining unknown constants, namely $C_{\bar X}$ (see Eq. (\[eq:Xt\])), $C_{\bar Y}$ (Eq. (\[eq:Y0\])), and $CC_1$. The latter two constants appear in particular in Eqs. (\[eq:pf1\]), (\[eq:pf2\]), (\[eq:moments\]) and (\[eq:mu1\]).

\[sec:noncritical\] Ebert-Van Saarloos calculation and its extension
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The original calculation of Ebert and Van Saarloos aimed at finding properties of the solutions to the FKPP equation \_t (t,x)=\_x\^2(t,x)+(t,x)-\^2(t,x) \[eq:FKPPphi\] for $\phi\ll 1$, with a steep enough initial condition, e.g. $\phi(t=0,x)=\theta(-x)$. This equation is actually the same as Eq. (\[eq:FKPP\]), with the correspondence $\phi(t,x)=1-Q(t,x)$. The nonlinearity can essentially be viewed as a moving absorptive boundary on a linear partial differential equation, the position of the boundary being set in such a way that $\phi$ has a maximum at a fixed height.

To determine the value of the constants $C_{\bar X}$ and $C_{\bar Y}$, we 