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### Strong spatial mixing for lattice graphs with fewer colours

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UNSPECIFIED
(2004)
*Strong spatial mixing for lattice graphs with fewer colours.*
In: 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, ITALY, OCT 17-19, 2004. Published in: 45TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, PROCEEDINGS
pp. 562-571.

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## Abstract

Recursively-constructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to non-tree-like graphs such as the integer lattice. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some Delta greater than or equal to 3, the maximum degree of G is at most Delta. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > alphaDelta, where alpha approximate to 1.76322 is the solution to alpha(alpha) = e. Note that we have no additional lower bound on q or Delta. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics such as the integer lattice Z(d) and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighbourhood of a vertex grows sub-exponentially in k), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. We extend our general result, obtaining, for example, the first "handproof" of strong spatial mixing for 7-colourings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal (for the rectangular lattice) and by Bubley, Dyer Greenhill and Jerrum). The extension also gives the first hand proof of strong spatial mixing for 5-colourings of triangle-free 3-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal.) Towards the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for q = 10 on the lattice Z(3). Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us the first (machine-assisted) proof of strong spatial mixing for 10-colourings of the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.).

Item Type: | Conference Item (UNSPECIFIED) |
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Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |

Series Name: | ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE |

Journal or Publication Title: | 45TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, PROCEEDINGS |

Publisher: | IEEE COMPUTER SOC |

ISBN: | 0-7695-2228-9 |

ISSN: | 0272-5428 |

Official Date: | 2004 |

Number of Pages: | 10 |

Page Range: | pp. 562-571 |

Publication Status: | Published |

Title of Event: | 45th Annual IEEE Symposium on Foundations of Computer Science |

Location of Event: | Rome, ITALY |

Date(s) of Event: | OCT 17-19, 2004 |

URI: | http://wrap.warwick.ac.uk/id/eprint/34137 |

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