Cutoff for a class of auto-regressive models with vanishing additive noise

We analyze the convergence rates for a family of auto-regressive Markov chains $(X_k^{(n)}{)}_{k\ge 0}$ on ${\mathbb{R}}^d$, where at each step a randomly chosen coordinate is replaced by a noisy damped weighted average of the others. The interest in the model comes from the connection with a certain Bayesian scheme introduced by de Finetti in the analysis of partially exchangeable data. Our main result shows that, when $n$ gets large (corresponding to a vanishing noise), a cutoff phenomenon occurs.