On the cut-off phenomenon for the transitivity of randomly generated subgroups

Abstract

Consider K ≥ 2 independent copies of the random walk on the symmetric group SN starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $n\in \mathbb{N}$ \end{document} **image** , let Gn be the subgroup of SN generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut-off phenomenon at time N ln(N)/(2K) for the non-existence of fixed point of Gn and for the transitivity of Gn, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non-existence of a fixed point of Gn appears at time of order \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $N^{1+\frac{2}{K}}$ \end{document} **image** (at least for K ≥ 3), but there is no cut-off phenomenon. In the latter model, we recover a cut-off phenomenon for the non-existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.