Left and right convergence of graphs with bounded degree

Christian Borgs, Jennifer Chayes, Jeff Kahn, L. Lovász

Research output: Contribution to journalArticle

32 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalRandom Structures and Algorithms
Volume42
Issue number1
DOIs
Publication statusPublished - Jan 2013

Fingerprint

Physics
Statistical Physics
Graph in graph theory
Homomorphisms
Counting
Mean Convergence
Cluster Expansion
Convergent Sequence
Partition Function
Extremes
Uniqueness
Equivalence
Converge
Imply
Model

Keywords

  • Cluster expansion
  • Dobrushin Uniqueness
  • Graph limits
  • Left convergence
  • Right convergence

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Software
  • Mathematics(all)
  • Applied Mathematics

Cite this

Left and right convergence of graphs with bounded degree. / Borgs, Christian; Chayes, Jennifer; Kahn, Jeff; Lovász, L.

In: Random Structures and Algorithms, Vol. 42, No. 1, 01.2013, p. 1-28.

Research output: Contribution to journalArticle

Borgs, Christian ; Chayes, Jennifer ; Kahn, Jeff ; Lovász, L. / Left and right convergence of graphs with bounded degree. In: Random Structures and Algorithms. 2013 ; Vol. 42, No. 1. pp. 1-28.
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