Using High Level Probability Theory (eg Markov Chain Mixing) in Cryptography/Cryptanalysis
I think you're being too dismissive and thinking of this as a "side project".

The challenge is representing the action of the cryptographic mappings such as the key schedule and the round functions which result in a pseudorandom permutation that can only sample a vanishingly small subset (a fraction $$\frac{2^k}{(2^n)!}$$ for keylength $k$ block length $n$, which are usually the same, but not necessarily) of all permutations in $S_{2^n}$.

The key is random, but the subkeys are not, they are directly derived from the key, so you have $k$ bit maximum key entropy used typically for $kr$ total subkeys bits.

More importantly, you need a worst-case approach while to my understanding the mixing time convergence to uniform is more of an average case criterion.

Are you aware of work by Lai-Massey-Murphy, Luby Rackoff. You can start by tracking the references to those.

And poly time mixing may tell you hundreds of rounds are enough on average but you'll want to use sublinear number of rounds. 

This answer focused on block ciphers alone.