We investigate the convergence towards equilibrium of the noisy voter model, evolving in the complete graph with n vertices. The noisy voter model is a version of the voter model, on which individuals change their opinions randomly due to external noise. Specifically, we determine the profile of convergence, in Kantorovich distance (also known as 1-Wasserstein distance), which corresponds to the Kantorovich distance between the marginals of a Wright-Fisher diffusion and its stationary measure. In particular, we demonstrate that the model does not exhibit cut-off under natural noise intensity conditions. In addition, we study the time the model needs to forget the initial location of particles, which we interpret as the Kantorovich distance between the laws of the model with particles in fixed initial positions and in positions chosen uniformly at random. We call this process thermalization and we show that thermalization does exhibit a cut-off profile. Our approach relies on Stein's method and analytical tools from PDE theory, which may be of independent interest for the quantitative study of observables of Markov chains.
