  To a subshift over a finite alphabet, one can naturally associate an infinite
family of finite graphs, called its Rauzy graphs. We show that for a subshift
of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$
in the sense of Benjamini-Schramm convergence if and only if its complexity
function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We
then apply this criterion to many examples of well-studied dynamical systems.
If the subshift is moreover uniquely ergodic then we show that the limit of
labelled Rauzy graphs if it exists can be identified with the unique invariant
measure. In addition we consider an example of a non uniquely ergodic system
recently studied by Cassaigne and Kabor\'e and identify a continuum of
invariant measures with subsequential limits of labelled Rauzy graphs.
