Abstract: Create continuous-time random walk on the symmetric group by successively composing independent transpositions chosen uniformly at random from among the possibilities, at rate n/2.  The uniform distribution is stationary for this Markov chain. A well-known result of Schramm states that this process undergoes a phase transition: if t < 1, the cycles of the permutation at time t are O(log n) in size, whereas for t > 1, a positive proportion of the numbers {1,2,\ldots,n} are contained in giant cycles, whose relative sizes are distributed approximately as Poisson-Dirichlet(0,1) (so that although the whole random walk is far from having reached stationarity, it has mixed on part of the space).  In this talk, I will characterise the behaviour of the critical random transposition random walk, and shed light on the emergence of the Poisson-Dirichlet distribution.  This is joint work with Dominic Yeo.