Refined estimates for some basic random walks on the symmetric and alternating groups

Details Refined estimates for some basic random walks on the symmetric and alternating groups Refined estimates for some basic random walks on the symmetric and alternating groups

2008-09-03

arxiv.org/abs/0809.0688v1

Mathematics

Probability

Accepted by Latin American Journal of Probability and Mathematical Statistics (ALEA)

Scientific paper

We give refined estimates for the discrete time and continuous time versions of some basic random walks on the symmetric and alternating groups $S_n$ and $A_n$. We consider the following models: random transposition, transpose top with random, random insertion, and walks generated by the uniform measure on a conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by the uniform measure on a conjugacy class, we show that in continuous time the $\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result, along with the results of M\"uller, Schlage-Puchta and Roichman, demonstrates that the continuous time version of these walks may take much longer to reach stationarity than its discrete time counterpart.

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