Markov chain on regular polyhedra

Abstract

The following modification of G.LETAC and L.TAKACS`` probabilistic model (1980) is considered : A bug runs along the edges of a regular dodecahedron D with constant speed : One edge per unit of time. At time 0 the bug is on some vertex A; assume that the bug comes to a vertex, it chooses with probability 1/5 the edge it passed, with equal probability 2/5 one of the other two edges. In this thesis, we first describe the above model by Markov chain and compute explicitly the probability u$_n$ that the bug is on A at time n in terms of n using the concept of quotient Markov chain. We also compute the probability u$_n$ for other solids.