Rates of convergence for Gibbs sampling in the analysis of almost exchangeable data

Motivated by de Finetti's representation theorem for partially exchangeable arrays, we want to sample $\mathbf p \in [0,1]^d$ from a distribution with density proportional to $\exp(-A^2\sum_{i<j}c_{ij}(p_i-p_j)^2)$. We are particularly interested in the case of an almost exchangeable array ($A$ large). We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every fixed matrix $C=(c_{ij})$, and large enough $A$, mixing happens in $\Theta(A^2)$ steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix $C$ through few relevant spectral parameters.