The random transposition walk on the symmetric group 
is the random walk which, at every step, multiplies
the current permutation by a uniformly chosen transposition. Thus its one-step probability
measure is
 |
(1)
|
The transposition-only walk alternates between even permutations and odd permutations. The aperiodic
version allowing identity permutation steps
has a cutoff in total variation distance
at 
steps with a window of order 
(Diaconis and Shahshahani 1981), with a parity-adjusted analogue
for the transposition-only walk. At the cutoff scale, the fixed
point count approaches a Poisson distribution.
Arcona (2026) uses group representation theory
to determine the limiting distributions of the numbers of j
for every fixed 
, and also treats the star-transposition
walk in which each step swaps 1 with a uniformly chosen other point.
