Diaconis mixing-time conjecture for S_n

Prove that the mixing time of random walks on S_n is O(n^3 log(n)) for every choice of generators.

Background

The second broad conjecture recorded in the paper concerns random walks on symmetric groups. It predicts a uniform O(n3 log(n)) upper bound on mixing time over all generating sets, complementing the quadratic diameter conjecture.

References

In the case of the symmetric group $S_n$ there are two open conjectures which are easy to formulate, but somewhat representative for the field.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 2.6, “Cayley graphs: applications, diameters, random walks, open conjectures”