### 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 |
---|---|

Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Random Structures and Algorithms |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2013 |

### Fingerprint

### 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

*Random Structures and Algorithms*,

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

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

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 42, no. 1, pp. 1-28. https://doi.org/10.1002/rsa.20414

}

TY - JOUR

T1 - Left and right convergence of graphs with bounded degree

AU - Borgs, Christian

AU - Chayes, Jennifer

AU - Kahn, Jeff

AU - Lovász, L.

PY - 2013/1

Y1 - 2013/1

N2 - 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.

AB - 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.

KW - Cluster expansion

KW - Dobrushin Uniqueness

KW - Graph limits

KW - Left convergence

KW - Right convergence

UR - http://www.scopus.com/inward/record.url?scp=84869877581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869877581&partnerID=8YFLogxK

U2 - 10.1002/rsa.20414

DO - 10.1002/rsa.20414

M3 - Article

AN - SCOPUS:84869877581

VL - 42

SP - 1

EP - 28

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -