  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 function. Applications include the
uniform infinite planar quadrangulation and long-range percolation clusters.
