Emergence of giant cycles and slowdown transition in random transpositions and k-cycles

Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly: the acceleration (i.e., the second time derivative of the distance) drops from 0 to -infinity at this time as n -> infinity. On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is considerably simpler and holds more generally than in a previous result of Oded Schramm [19] for random transpositions. It turns out that in the case of random k-cycles, this critical time is proportional to 1/[k(k - 1)], whereas the mixing time is known to be proportional to 1/k.