  We study various classes of random processes defined on the regular tree
$T_d$ that are invariant under the automorphism group of $T_d$. Most important
ones are factor of i.i.d. processes (randomized local algorithms), branching
Markov chains and a new class that we call typical processes. Using Glauber
dynamics on processes we give a sufficient condition for a branching Markov
chain to be factor of i.i.d. Typical processes are defined in a way that they
create a correspondence principle between random $d$-reguar graphs and ergodic
theory on $T_d$. Using this correspondence principle together with entropy
inequalities for typical processes we prove a family of combinatorial
statements about random $d$-regular graphs.
