  Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$
is fixed and $n$ goes to $\infty$ then we can relate typical (large
probability) properties of such random graph to a family of invariant random
processes (called "typical" processes) on the infinite $d$-regular tree $T_d$.
This correspondence between ergodic theory on $T_d$ and random regular graphs
is already proven to be fruitful in both directions. This paper continues the
investigation of typical processes with a special emphasis on entropy. We study
a natural notion of micro-state entropy for invariant processes on $T_d$. It
serves as a quantitative refinement of the notion of typicality and is tightly
connected to the asymptotic free energy in statistical physics. Using entropy
inequalities, we provide new sufficient conditions for typicality for edge
Markov processes. We also extend these notions and results to processes on
unimodular Galton-Watson random trees.
