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

André Galligo (1) , Laurent Miclo (2)

1 GALAAD - Geometry, algebra, algorithms
2 IMT - Institut de Mathématiques de Toulouse UMR5219

Consider $K\geq2$ independent copies of the random walk on the symmetric group $S_N$ starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in\NN$, let $G_n$ be the subgroup of $S_N$ 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 $G_n$ and for the transitivity of $G_n$, 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 $G_n$ appears at time of order $N^{1+\frac 2K}$ (at least for $K\geq3$), 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.

Keywords: Isoperimetric bounds; Random walks on symmetric groups; Uniform transposition model; Uniform successive transposition model; Cut-off phenomenon; Flared transition phenomenon; Existence of fixed point(s) for random subgroups; Transitivity for random subgroups; Spectral analysis; Coupling techniques; Exclusion processes; Isoperimetric bounds.

Domains: Probability [math.PR]

André Galligo, Laurent Miclo. On the cut-off phenomenon for the transitivity of randomly generated subgroups. Random Structures and Algorithms, 2012, 40 (2), pp.189-219. ⟨hal-00384188v2⟩