  We show that the local weak limit of a sequence of finite expander graphs
with uniformly bounded degree is an ergodic (or extremal) unimodular random
graph. In particular, the critical probability of percolation of the limiting
random graph is constant almost surely. As a corollary, we obtain an
improvement to a theorem by Benjamini-Nachmias-Peres (2011) on locality of
percolation probability for finite expander graphs with uniformly bounded
degree where we can drop the assumption that the limit is a single rooted
graph.
