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Strong spatial mixing with fewer colors for lattice graphs
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UNSPECIFIED (2005) Strong spatial mixing with fewer colors for lattice graphs. SIAM JOURNAL ON COMPUTING, 35 (2). pp. 486517. doi:10.1137/S0097539704445470
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Official URL: http://dx.doi.org/10.1137/S0097539704445470
Abstract
Recursivelyconstructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontreelike graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a trianglefree graph and that for some. Delta >= 3, the maximum degree of G is at most Delta. We show that the spin system consisting of q colorings of G has strong spatial mixing, provided q > alpha Delta gamma, where alpha approximate to 1.76322 is the solution to alpha(alpha) = e, and gamma = 4 alpha(3) 6 alpha(2) 3 alpha+4/ 2(alpha(2) 1) approximate to 0.47031. 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 neighborhood 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. Our general result gives, for example, a "hand proof" of strong spatial mixing for 7colorings of trianglefree 4regular graphs. (Computerassisted proofs of this result were provided by Salas and Sokal [ J. Stat. Phys., 86 ( 1997), pp. 551  579] ( for the rectangular lattice) and by Bubley, Dyer, Greenhill, and Jerrum [SIAM J. Comput., 29 ( 1999), pp. 387  400].) It also gives a hand proof of strong spatial mixing for 5colorings of triangle free 3regular graphs. ( A computerassisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal [ J. Stat. Phys., 86 ( 1997), pp. 551  579].) Toward 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 a ( machineassisted) proof of strong spatial mixing for 10colorings of the triangular lattice. ( Such a proof for 11 colors was given by Salas and Sokal [ J. Stat. Phys., 86 ( 1997), pp. 551  579].) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colorings of neighborhoodamenable graphs. ( It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for Z(d).) Thus our strong spatial mixing results give rapid mixing corollaries for neighborhoodamenable graphs such as lattices.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software Q Science > QA Mathematics 

Journal or Publication Title:  SIAM JOURNAL ON COMPUTING  
Publisher:  SIAM PUBLICATIONS  
ISSN:  00975397  
Official Date:  2005  
Dates: 


Volume:  35  
Number:  2  
Number of Pages:  32  
Page Range:  pp. 486517  
DOI:  10.1137/S0097539704445470  
Publication Status:  Published 
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