Reference Request: soundness in a ZKP achieved by walking along a doubly-stochastic Markov chain?
This seems to be explicit or at least implicit in Lutomirski's 2011 "Component mixers and a hardness result for counterfeiting quantum money" - arXiv

Lutomirski formally defines component mixers as a family $\{M_i\}$ of one-to-one maps between elements of a set $S$ partitioned into $\{S_1,S_2,\cdots,S_c\}$.  

He gives a number of example $S$'s, such as the set of all graphs on $n$ vertices, where each partition $S_1,S_2,\cdots,S_c$ corresponds to different isomorphism classes of graphs.  Each element $g\in S$ is a given graph on $n$ vertices, and is a member of only one partition $S_g$.  "Mixing" means to randomly draw another element $g'$ from the same $S_g$.

He notes that for many sets $S$ one can mix easily enough, with permutations or straight-line-programs.  Otherwise, he comments that:


  "In general, given a Markov chain over $S$ that does not mix between components but mixes rapidly over each component, each step of which consists of choosing uniform random sample from a set of invertible rules and applying that rule, then iterating that Markov chain to amplify its spectral gap will give a component mixer."


Lutomirski goes on to provide a $\mathsf{coAM}$ protocol for determining whether the number of components (partitions) is greater than $1$.