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Journal of Theoretical Probability

, Volume 30, Issue 2, pp 639–654 | Cite as

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

  • Malin P. ForsströmEmail author
  • Johan Jonasson
Article

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 \( L^2 \) 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 

Mathematics Subject Classification (2010)

60J10 05C81 

Notes

Acknowledgments

This research was supported by Chalmers University of Technology and the Knut and Alice Wallenberg foundation.

Compliance with Ethical Standards

Conflict of interest

None.

References

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and the University of GothenburgGöteborgSweden

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