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

TL;DR

This paper investigates phase transitions in random walks on permutation groups, revealing a critical time where the process experiences a slowdown and the emergence of giant cycles, with results applicable to k-cycles.

Contribution

It demonstrates a unified phase transition in random conjugacy class walks, simplifying previous proofs and extending results to general k-cycles.

Findings

Critical time for phase transition proportional to 1/[k(k-1)]

Giant cycles emerge at the critical time

Random walk slowdown occurs simultaneously with cycle growth

Abstract

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 (i.e., the acceleration drops from 0 to -\infty at this time as n tends to \infty). On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is both considerably simpler and more general than in a previous result of Oded Schramm (2005) 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.