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

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

2010-04-20

arxiv.org/abs/1004.3530v1

Mathematics

Probability

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-474898