than 2, and up to an additional constant for the first moment. Throughout, we shall aim at the accuracy ${\cal O}\left(1/\sqrt{t_0},1/\sqrt{t},1/\sqrt{t-t_0}\right)$ for $p(f)$ and neglect higher powers of these expansion variables.

\[sec:distrib1\]Effect of a fluctuation on $\ln Y$
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We first compute $\bar Y_t$, namely the variable $Y_t$ in the mean-field approximation with the cutoff in the tail. Using the definition of the variable $Y_t$ in Sec. \[sec:var\] and using Eqs. (\[eq:shape\]),(\[eq:Xt\]), we find $$\bar Y_t=\sqrt{t}\int_{-\infty}^{+\infty}dx\, \psi_{\bar X_t}(x,t)e^{x-2t}=-2\alpha
\left(1+\frac{C_{\bar Y}}{\sqrt{t}}\right),
\label{eq:Y0}$$ at first order in $1/\sqrt{t}$. $C_{\bar Y}$ is a constant literally equal to $C_{\bar X}-\frac{\beta\sqrt{\pi}}{2\alpha}$ in this calculation, but there would also be other contributions to $C_{\bar Y}$ that we cannot get from the large-$t$ asymptotic shape of the front exhibited in Eq. (\[eq:shape\]). We shall postpone the full calculation of $C_{\bar Y}$ to Sec. \[sec:det\].

It turns out that the term of order $1/\sqrt{t}$ in Eq. (\[eq:Y0\]) generates ${\cal O}(1/t)$ contributions to the distributions and to the moments that we shall address. Hence it is enough for our purpose to keep no more than the constant term, namely we write |Y\_t-2. \[eq:Y00\] We perform a more complete calculation in Appendix \[sec:details\], keeping the subleading terms, in order to 