The Spectrum and Convergence Rates of Exclusion and Interchange Processes on the Complete Graph
We give a short and completely elementary method to find the full spectrum of the exclusion process and a nicely limited superset of the spectrum of the interchange process (a.k.a. random transpositions) on the complete graph. In the case of the exclusion process, this gives a simple closed-form expression for all the eigenvalues and their multiplicities. This result is then used to give an exact expression for the distance in \( L^2 \) from stationarity at any time and upper and lower bounds on the convergence rate for the exclusion process. In the case of the interchange process, upper and lower bounds are similarly found. Our results strengthen or reprove many known results about the mixing time for the two processes in a very simple way.
KeywordsRandom transpositions Exclusion process Interchange process Mixing time Spectrum
Mathematics Subject Classification (2010)60J10 05C81
This research was supported by Chalmers University of Technology and the Knut and Alice Wallenberg foundation.
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