Has it converged?

Two chains are started from opposite extremes: one from a uniformly random field (hot), one from a single uniform label (cold). They run the same kernel, the same temperature and the same number of sweeps. Energy per site is bounded between them, so wherever the two curves meet, the chain has forgotten where it started.

None of this is new. Ordered and disordered starts, plotted per sweep and compared, are the ordinary equilibration check of computational statistical physics, set out in the standard texts for q-state Potts models by name (Berg, Introduction to Markov Chain Monte Carlo Simulations and their Statistical Analysis, 2004). It is here because in 1999 Efros and Leung set Gibbs sampling aside for texture partly because “it is not possible to assess when it has converged”: an obstacle that the field one shelf over had long treated as a first-week exercise.

The gap between the curves is a bound, not an estimate: any statistic that is monotone in the initial order sits between the two chains, so agreement is evidence of equilibration rather than a guess at it. It is a necessary condition and not a sufficient one, since two chains can both be stuck in the same place. Near the ordering transition the meeting point moves out sharply, which is critical slowing down, and it is the honest reason a fixed sweep budget cannot be quoted as a converged sample at every temperature.