| Título: | Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles |
| Autores: | Berestycki, Nathanael; University of Cambridge |
| Fecha: | 2011-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
Random permutations 60J10; 60K35; 60B15; 05C80; 05C12; 05C65 |
| Descripción: | 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 $1/k$. |
| Idioma: | No aplica |
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