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

Rates of convergence to equilibrium for Potlatch and Smoothing processes