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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 1719, 2004. Published in: 45TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, PROCEEDINGS pp. 562571. ISBN 0769522289. ISSN 02725428.
Research output not available from this repository, contact author.Abstract
Recursivelyconstructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to nontreelike graphs such as the integer lattice. Using this method, we obtain the following general result. Suppose that G is a trianglefree 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 qcolourings 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 distancek neighbourhood of a vertex grows subexponentially in k), strong spatial mixing implies that there is a unique infinitevolume 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 7colourings of trianglefree 4regular graphs. (Computerassisted 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 5colourings of trianglefree 3regular graphs. (A computerassisted 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 (machineassisted) proof of strong spatial mixing for 10colourings of the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.).
Item Type:  Conference Item (UNSPECIFIED)  

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:  0769522289  
ISSN:  02725428  
Official Date:  2004  
Dates: 


Number of Pages:  10  
Page Range:  pp. 562571  
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 1719, 2004 
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