a rich class of dynamics displaying cutoff. One emblematic example is the *stochastic Ising model* at high enough temperature, for which the cutoff phenomenon has been established successively on the complete graph (@levin2010glauber), on lattices (@ding2009mixing, @lubetzky2014cutoff), and finally on any sequence of graphs (@lubetzky2014universality). Other examples include the *Potts model* (@cuff2012glauber), the *East process* (@ganguly2013cutoff), or the *Simple Exclusion process* on the cycle (@lacoin2015cutoff).

The problem of singling out abstract conditions under which the cutoff phenomenon occurs, without necessarily pinpointing its precise location, has drawn considerable attention. In 2004, @peresamerican proposed a simple spectral criterion for reversible chains, known as the *product condition*. Although counter-examples have quickly been constructed (see @levin2009markov [Chapter 18] and @chen2008cutoff [Section 6]), the condition is widely believed to be sufficient for “most" chains. This has already been verified for birth-and-death chains (@ding2010total) and, more generally, for random walks on trees (@basucharacterization). The latter result relies on a promising characterization of cutoff in terms of the concentration of hitting times of “worst” (in some sense) sets. See also @oliveira2012mixing, @sousi2013hitting, @griffiths2014tight and @hermon2015technical.

Many natural families of Markov chains are now believed to exhibit cutoff. Yet, establishing this phenomenon rigorously requires a very detailed understanding 