Rates of convergence to equilibrium for Potlatch and Smoothing processes

Abstract

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 Z d . 2010 Mathematics Subject Classification. 60K35.

Ergodicity and steady state analysis for Interference Queueing Networks

“…Moreover, the ratio-type functional dependence of the service rates on neighboring queues makes obtaining quantitative estimates challenging and most of the analysis necessarily has to rely on 'soft' arguments using qualitative traits of the model. Recently, motivated by this model, the first author revisited an interacting particle system called the Potlatch process [4], which shares many aspects in common with this model, but the simpler functional form of rates enables one (see [3]) to quantify rates of convergence (locally and globally) to equilibrium. Similar models have also appeared in the economics literature to analyze opinion dynamics on social networks [1].…”