NDLI: Strong spatial mixing for lattice graphs with fewer colours

Content Provider	IEEE Xplore Digital Library
Author	Goldberg, L.A. Martin, R. Paterson, M.
Copyright Year	2004
Description	Author affiliation: Dept. of Comput. Sci., Warwick Univ., Coventry, UK (Goldberg, L.A.; Martin, R.; Paterson, M.)
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 nontree-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 /spl Delta/ /spl ges/ 3, the maximum degree of G is at most /spl Delta/. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > /spl alpha//spl Delta/, where /spl alpha/ /spl ap/ 1.76322 is the solution to /spl alpha//sup /spl alpha// = e. Note that we have no additional lower bound on q or /spl 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 /spl Zopf//sup 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 "hand proof" 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 (1997) for the rectangular lattice and by Bubley et al. (1999)). 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 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 /spl Zopf//sup 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.).
Sponsorship	IEEE Comput. Soc. Tech. Soc. on Foundations of Comput
Starting Page	562
Ending Page	571
File Size	416336
Page Count	10
File Format	PDF
ISBN	0769522289
ISSN	02725428
DOI	10.1109/FOCS.2004.63
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2004-10-17
Publisher Place	Italy
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Lattices Computer science Tree graphs Physics Computational geometry Antiferromagnetic materials
Content Type	Text
Resource Type	Article

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