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Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles

   
@article{EJP850,
	author = {Nathanael Berestycki},
	title = {Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {16},
	year = {2011},
	keywords = {Random permutations},
	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: the acceleration (i.e., the second time derivative of the distance) drops from $0$ to $-\infty$ at this time as $n\to\infty$. 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  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 $1/k$.},
	pages = {no. 5, 152-173},
	issn = {1083-6489},
	doi = {10.1214/EJP.v16-850},    
        url = {http://ejp.ejpecp.org/article/view/850}}