and }& {\ensuremath{\mathbb{E}}}\left[e^{\theta D}\right]<\infty\textrm{ for some }\theta>0,
\end{array}$$ ensures that the conditions (\[assume:Delta\]), (\[assume:delta\]) and (\[assume:sigma\]) hold almost surely. Thus, Theorem \[thm:main\] applies with the parameters $\mu,\sigma$ and $N$ now being random. But the latter clearly concentrate around their deterministic counterparts, in the following sense: $$\begin{aligned}
N&=& n{\ensuremath{\mathbb{E}}}[D]+O_{\ensuremath{\mathbb{P}}}\left(n^{\frac 12}\right)\\
\mu&=& \mu_\star+O_{\ensuremath{\mathbb{P}}}\left(n^{-\frac 12}\right)\ \textrm{ with }\ \mu_\star={{\ensuremath{\mathbb{E}}}[D\log(D-1)]}/{{\ensuremath{\mathbb{E}}}[D]}\\
\sigma&=&\sigma_\star+O_{\ensuremath{\mathbb{P}}}\left(n^{-\frac 12}\right)\ \textrm{ with }\ \sigma_\star^2={{\ensuremath{\mathbb{E}}}\left[D\left\{\log(D-1)-\mu_\star\right\}^2\right]}/{{\ensuremath{\mathbb{E}}}[D]}. \end{aligned}$$ Those error terms are small enough to allow one to substitute $n,\mu_\star,\sigma_\star$ for $N,\mu,\sigma$ without affecting the convergence stated in Theorem \[thm:main\].

Related work.
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The first instances of the cutoff phenomenon were discovered in the early 80’s by @diaconis1981generating and @aldous1983mixing, in the context of card shuffling: given a certain procedure for shuffling a deck of cards, there exists a quite precise number of shuffles slightly below which the deck is far from being mixed, and slightly above which it is almost completely mixed. The term *cutoff* and the general formalization appeared shortly after, in the seminal paper by @aldous1986shuffling. Since then, this remarkable behaviour has been identified in a variety of other contexts, see e.g., @diaconis1996cutoff, @chen2008cutoff, or the survey by @saloff2004random for random walks on finite groups.

Interacting particle systems in statistical mechanics provide 