Left and right convergence of graphs with bounded degree

Authors:
Christian Borgs;Jennifer Chayes;Jeff Kahn;László Lovász
Affiliations:
Microsoft Research New England, Cambridge, Massachusetts 02142;University of Cambridge, Cambridge, UK and University of Memphis, Memphis, Tennessee and Microsoft Research, Redmond, Washington and University of Oxford, Oxford, UK;Department of Mathematics, Rutgers University, Piscataway, New Jersey;Microsoft Research, Redmond, Washington and Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary
Venue:
Random Structures & Algorithms
Year:
2013

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Abstract

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 (Supported by OTKA (67867) and ERC (227701).)