Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 205216
author:  Mikhail Menshikov, Dimitri Petritis and Serguei Popov 

title:  Bindweeds or random walks in random environments on multiplexed trees and their asympotics 
keywords:  Markov chain, trees, random environment, recurrence criteria, matrix multiplicative cascades 
abstract: 
We report on the asymptotic behaviour of a new model of
random walk, we term the bindweed model, evolving in a
random environment on an infinite multiplexed tree. The
term multiplexed means that the model can be viewed as a
nearest neighbours random walk on a tree whose vertices
carry an internal degree of freedom from the finite set
{1,...,d}
, for some integer
d
. The consequence of the internal degree of freedom
is an enhancement of the tree graph structure induced by
the replacement of ordinary edges by multiedges, indexed
by the set
{1,...,d} × {1,...,d}
. This indexing conveys the information on the
internal degree of freedom of the vertices contiguous to
each edge. The term random environment means that the
jumping rates for the random walk are a family of
edgeindexed random variables, independent of the natural
filtration generated by the random variables entering in
the definition of the random walk; their joint distribution
depends on the index of each component of the multiedges.
We study the large time asymptotic behaviour of this random
walk and classify it with respect to positive recurrence or
transience in terms of a specific parameter of the
probability distribution of the jump rates. This
classifying parameter is shown to coincide with the
critical value of a matrixvalued multiplicative cascade on
the ordinary tree (i.e. the one without internal degrees of
freedom attached to the vertices) having the same vertex
set as the state space of the random walk. Only results are
presented here since the detailed proofs will appear
elsewhere.

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reference:  Mikhail Menshikov and Dimitri Petritis and Serguei Popov (2003), Bindweeds or random walks in random environments on multiplexed trees and their asympotics, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 205216 
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