Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique

The main purpose of this paper is to exhibit the cutoff phenomenon, studied by Aldous and Diaconis [AD]. Let ${\displaystyle Q^{\cdot k}}$ denote a transition kernel after k steps and π be a stationary measure. We have to find a critical value ${\displaystyle k_n}$ for which the total variation norm between ${\displaystyle Q^{\cdot k}}$ and π stays very close to 1 for ${\displaystyle k\ll k_n}$, and falls rapidly to a value close to 0 for ${\displaystyle k\ge k_n}$ with a fall-off phase much shorter than ${\displaystyle k_n}$. According to the work of Diaconis and Shahshahani [DS], one can naturally conjecture, for a conjugacy class with n-c fixed points, with ${\displaystyle c\ll n}$, that the associated random walk presents a cutoff, with critical value equal to (1/c)nln(n). Using Fourier analysis, we prove that, in this context, the critical value can not be less than (1/c)nln(n). We also prove that the conjecture is true for conjugacy classes with at least n-6 fixed points and for a conjugacy class of cycle length 7.