  We study random walks in a random environment on a regular, rooted, coloured
tree. The asymptotic behaviour of the walks is classified for
ergodicity/transience in terms of the geometric properties of the matrix
describing the random environment. A related problem, with only one type of
vertices and quite stringent conditions on the transition probabilities but on
general trees has been considered previously in the literature LyoPem. In the
presentation we give here, we restrict the study of the process on a regular
graph instead of the irregular graph used in LyoPem. The close connection
between various problems on random walks in random environment and the so
called multiplicative chaos martingale is underlined by showing that the
classification of the random walk problem can be drawn by the corresponding
classification for the multiplicative chaos, at least for those situations
where both problems have been solved by independent methods. The chaos
counterpart of the problem we considered here has not yet been solved. The
results we obtain for the random walk problem localise the position of the
critical point. We conjecture that the additional conditions needed for the
chaos problem to have non trivial solutions will be the same as the ones needed
for the random walk to be null recurrent.
