Left and right convergence of graphs with bounded degree

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 (rig...

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Bibliographic Details
Published in:Random structures & algorithms 2013-01, Vol.42 (1), p.1-28
Main Authors: Borgs, Christian, Chayes, Jennifer, Kahn, Jeff, Lovász, László
Format: Article
Language:English
Subjects:
cluster expansion
Dobrushin Uniqueness
graph limits
left convergence
right convergence
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Summary: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, Lovász, Sós 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. © 2012 Wiley Periodicals, Inc. Random Struct. 2012
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20414