Journal
RANDOM STRUCTURES & ALGORITHMS
Volume 42, Issue 1, Pages 1-28Publisher
WILEY
DOI: 10.1002/rsa.20414
Keywords
cluster expansion; Dobrushin Uniqueness; graph limits; left convergence; right convergence
Funding
- NSF [DMS0701175]
- OTKA [67867]
- ERC [227701]
- European Research Council (ERC) [227701] Funding Source: European Research Council (ERC)
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The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovasz, Sos and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case, if the set of graphs in the definition of right-convergence is appropriately restricted. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left-convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness. (C) 2012 Wiley Periodicals, Inc. Random Struct. 2012
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