$f$ for large positive $f$. However, numerically, we do not find evidence for such a modification: It seems that Eq. (\[eq:pf1\]) has a more general validity. We do not have a good explanation for this surprising fact in the context of our phenomenological model for fluctuations. But it turns out that a different calculation of the positive $f$ fluctuations outlined in Appendix \[sec:exact\] does not have such limitations.

\[sec:mu1\] Correction to the first moment of $f$ due to the fluctuations
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Since it is not possible to use Eq. (\[eq:moments\]) to get the first moment of $f$, we shall arrive at its expression through a direct calculation. We must take into account the expansion (keeping terms at least as large as $1/\sqrt{t}$, $1/\sqrt{t_0}$, $1/\sqrt{t-t_0}$) of the density of particles in the deterministic limit with a discreteness cutoff, and, in addition, the effect of the fluctuations which intermittently speed up the evolution. We have already guessed that there is an ${\cal O}(1/\sqrt{t})$ contribution to the deterministic evolution (see Eq. (\[eq:Y0\])), but a full calculation will eventually be needed. Here, we shall simply denote by $C_{\bar Y}$ its coefficient.\
The average of $f=\ln Y_t-\ln Y_{t_0}$ over realizations has thus a mean-field contribution, 