Bindweeds or random walks in random environments on multiplexed trees and their asympotics
Résumé
We report on the asymptotic behaviour ofa new model of random walk, we term the bindweed model, evolving in a random environmenton an infinite multiplexed tree.The term \textit{multiplexed} means that the model can beviewed as a nearest neighbours random walk on a tree whosevertices carry an internal degree of freedom fromthe finite set $\{1,\ldots,d\}$, for some integer $d$.The consequence of the internal degree of freedom isan enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexedby the set $\{1,\ldots,d\}\times\{1,\ldots,d\}$. This indexingconveys the information on the internal degree of freedomof the vertices contiguous to each edge.The term \textit{random environment} means that the jumping ratesfor the random walk are a family of edge-indexed random variables, independentof the natural filtration generated by therandom variables entering in the definition of therandom walk; their joint distribution depends on the index of each componentof the multi-edges. We study the large time asymptotic behaviourof this random walk and classify it with respectto positive recurrence or transience in termsof a specific parameter of the probability distributionof the jump rates.This classifying parameter is shown to coincidewith the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{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 appearelsewhere.