The Spectrum and Convergence Rates of Exclusion and Interchange Processes on the Complete Graph

Malin P. Forsström () and Johan Jonasson ()
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Malin P. Forsström: Chalmers University of Technology and the University of Gothenburg
Johan Jonasson: Chalmers University of Technology and the University of Gothenburg

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 639-654

Abstract: Abstract 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 $$ L^2 $$ 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.

Keywords: Random transpositions; Exclusion process; Interchange process; Mixing time; Spectrum; 60J10; 05C81 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0660-6

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