  We analyze the local and global smoothing rates of the smoothing process and
obtain convergence rates to stationarity for the dual process known as the
potlatch process. For general finite graphs, we connect the smoothing and
convergence rates to the spectral gap of the associated Markov chain. We
perform a more detailed analysis of these processes on the torus. Polynomial
corrections to the smoothing rates are obtained. They show that local smoothing
happens faster than global smoothing. These polynomial rates translate to rates
of convergence to stationarity in $L^2$-Wasserstein distance for the potlatch
process on $\mathbb{Z}^d$.
