We present an improved version of Diaconis–Shahshahani upper bound lemma, which is used to compute the limiting value of the distance to stationarity. We then apply it to the random transposition shuffle.
"Limit profile for random transpositions." Ann. Probab. 48 (5) 2323 - 2343, September 2020. https://doi.org/10.1214/20-AOP1424