I don't mean to imply d(n) does not drop off exponentially (as it should), only that it has a sharp drop-off before the exponential scaling sets in, rather than

I don't mean to imply d(n) does not drop off exponentially (as it should), only that it has a sharp drop-off before the exponential scaling sets in, rather than a steady exponential all the way. Also, if you equip an Ising model with Glauber dynamics, the resulting Markov chain does exhibit a cutoff phenonmenon. See, e.g., https://arxiv.org/abs/0909.4320 https://arxiv.org/abs/0909.4320 .