Left and right convergence of graphs with bounded degree

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.