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Markov chain algorithms for Eulerian orientations and 3-colourings of 2-dimensional Cartesian grids

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  • Fehrenbach Johannes
  • Rüschendorf Ludger

Abstract

In this paper we establish that the natural single point update Markov chain (also known as Glauber dynamics) for counting the number of Euler orientations of 2-dimensional Cartesian grids is rapidly mixing. This extends a result of Luby, Randall, and Sinclair (2001) who consider the case where orientations in the boundary are fixed. Similarly, we also obtain a rapid mixing result for the 3-colouring of rectangular Cartesian grids without fixing the boundaries. The proof uses path coupling and comparison to related Markov chains which allow additional transitions and which can be analysed directly.

Suggested Citation

  • Fehrenbach Johannes & Rüschendorf Ludger, 2004. "Markov chain algorithms for Eulerian orientations and 3-colourings of 2-dimensional Cartesian grids," Statistics & Risk Modeling, De Gruyter, vol. 22(2/2004), pages 109-130, February.
    • Handle: RePEc:bpj:strimo:v:22:y:2004:i:2/2004:p:109-130:n:2
    DOI: 10.1524/stnd.22.2.109.49126
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