5. Chains on $S_n$

Cutoff for randomtorandom shuffle
Conjecture 5.1.
[Evita Nestoridi] The randomtorandom card shuffle on $S_n$ has cutoff at $\frac {3n}4 \log n$.
In separation distance there is an $n\log n$ upper bound [Nesterodi, White]. 
Random biased transpositions
The random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.
The random nearlyuniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.Problem 5.2.
[Igor Pak] Does the random nearlyuniform transposition walk on $S_n$ exhibit cutoff?
What about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\leq i,j\leq n$? 
Cutoff profile of random transpositions
Problem 5.3.
[NathanaĆ«l Berestycki] Can we obtain the cutoff profile for the uniform random transposition walk on $S_n$?
Cite this as: AimPL: Markov chain mixing times, available at http://aimpl.org/markovmixing.