 01469 Mikhail Menshikov, Dimitri Petritis
 Random walks in random environment on trees
and multiplicative chaos
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Dec 15, 01

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Abstract. 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 \cite{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 \cite{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.
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