### 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 language | English |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Random Structures and Algorithms |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2013 |

### Keywords

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

### ASJC Scopus subject areas

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

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## Cite this

*Random Structures and Algorithms*,

*42*(1), 1-28. https://doi.org/10.1002/rsa.20414