  Peres and Winkler proved a "censoring" inequality for Glauber dynamics on
monotone spins systems such as the Ising model. Specifically, if, starting from
a constant-spin configuration, the spins are updated at some sequence of sites,
then inserting another site into this sequence brings the resulting
configuration closer in total variation to the stationary distribution. We show
by means of simple counterexamples that the analogous statements fail for
Glauber dynamics on proper colorings of a graph, and for lazy transpositions on
permutations, answering two questions of Peres. It is not known whether the
censoring property holds in other natural settings such as the Potts model.
