Brunet-Derrida cutoff [@BD:1997], and it was shown to correctly represent the leading effect of the noise on the position of the front in the context of the stochastic FKPP equation.

    From now on, we shall focus on the right boundary. (The right and the left halves of the BRW essentially decouple once the system has grown large enough). The large-time expression of the shape of the particle density near the right boundary reads, in such a model $$\psi_{\bar X_t}(x,t)=\left[\alpha(x-\bar X_t)+\beta\right] 
    \exp\left(\bar X_t-x-\frac{(\bar X_t-x)^2}{4t}\right)
    \theta(\bar X_t-x)
    \label{eq:shape}$$ in the region $1\ll\bar X_t-x\lesssim \sqrt{t}$, where $$\bar X_t=2t-\frac32\ln t+\frac{C_{\bar X}}{\sqrt{t}}
    \label{eq:Xt}$$ is, up to an uninteresting non-universal additive constant, the position of the tip of the front, namely of the right boundary. The Heaviside step function $\theta$ enforces the fact that the particle density is 0 to the right of $\bar X_t$. $C_{\bar X}$, $\alpha<0$ and $\beta$ are constants undetermined at this stage. $\psi$ is essentially a decreasing exponential supplemented with a Gaussian and a linear prefactors. The $t$-dependence enters explicitly as the width of the Gaussian, and implicitly through the position $\bar X_t$ of the absorptive boundary. (There are corrections to the shape of the front at order $1/\sqrt{t}$, namely to the 