A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic functions. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.
"Ergodic theory on stationary random graphs." Electron. J. Probab. 17 1 - 20, 2012. https://doi.org/10.1214/EJP.v17-2401