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This paper is concerned with sampling from the uniform distribution onH-colourings of the n-vertex path using systematic scan Markov chains. AnH-colouring of the n-vertex path is a homomorphism from the n-vertex path tosome fixed graph H. We show that systematic scan for H-colourings of then-vertex path mixes in O(log n) scans for any fixed H. This is a significantimprovement over the previous bound on the mixing time which was O(n^5) scans.Furthermore we show that for a slightly more restricted family of H (where anytwo vertices are connected by a 2-edge path) systematic scan also mixes inO(log n) scans for any scan order. Finally, for completeness, we show that arandom update Markov chain mixes in O(n log n) updates for any fixed H,improving the previous bound on the mixing time from O(n^5) updates.

On systematic scan for sampling H-colourings of the path