  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.
