Wreath products are a type of semidirect product. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.
"On random walks on wreath products." Ann. Probab. 30 (2) 948 - 977, April 2002. https://doi.org/10.1214/aop/1023481013