# Refined estimates for some basic random walks on the symmetric and alternating groups - Mathematics > Probability

Abstract: We give refined estimates for the discrete time and continuous time versionsof some basic random walks on the symmetric and alternating groups $S n$ and$A n$. We consider the following models: random transposition, transpose topwith random, random insertion, and walks generated by the uniform measure on aconjugacy class. In the case of random walks on $S n$ and $A n$ generated bythe 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 withthe results of M\-uller, Schlage-Puchta and Roichman, demonstrates that thecontinuous time version of these walks may take much longer to reachstationarity than its discrete time counterpart.

Author: L. Saloff-Coste, J. Zuniga

Source: https://arxiv.org/