Left and right convergence of graphs with bounded degree

TL;DR

This paper establishes the equivalence of left and right convergence for bounded degree graph sequences, extending known results from dense graphs and linking graph limits to statistical physics models.

Contribution

It proves the equivalence of two notions of graph convergence in the bounded degree case, using statistical physics techniques.

Findings

Left convergence implies right convergence for bounded degree graphs.

Partition functions of certain statistical physics models converge for left-convergent sequences.

The proof employs cluster expansion and Dobrushin Uniqueness methods.

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, Lov\'asz, S\'os and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case. 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.