In particular, [@AlDi86 Fig. 2] is quite similar to our Fig. \[fig:H-decay\] below.

3.  The resemblance of to the results of Aldous and Diaconis for finite Markov chains in [@AlDi86; @Di96], and in particular for card shuffling, is not a coincidence. The equation can be interpreted as the Kolmogorov forward equation for a Markov process. Exponential rates for related Markov process with exponential rates have been obtained by probabilistic methods; see [@BL] for an early study of this type. However, the approach in [@BL] relies on compactness arguments and does not yield explicit values for $c$ or $\lambda$. One difference between our results and those for finite Markov chains is that in our case, the initial relative entropy can be infinite. In card shuffling, starting form a perfectly ordered deck of cards, one starts from a state of maximal—but finite—relative entropy, and the waiting time for uniformization from this state dominates that of any other starting point. For this reason, the initial waiting time for finite Markov chains is a universal “worst case”, while this is impossible in our setting; our result must refer to ${\mathcal{E}}^d(\tilde f^I)$.

4.  Our bound on the decay rate is monotonically decreasing in $L$ and 