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) in  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.
This work was supported by the Loève Fellowship at UC Berkeley.
This work was completed when the author was a graduate student at UC Berkeley. The author thanks Nike Sun for suggesting this problem and for some helpful discussions. He also thanks Tom Hutchcroft for very helpful feedback on an earlier version of the paper. The author thanks the anonymous referee whose careful reading and detailed comments helped improve the paper.
"A note on the local weak limit of a sequence of expander graphs." Electron. Commun. Probab. 26 1 - 6, 2021. https://doi.org/10.1214/21-ECP402