181 research outputs found
Localization on 5 sites for vertex reinforced random walks: Towards a characterization
We continue the investigation of the localization phenomenon for a Vertex
Reinforced Random Walk on the integer lattice. We provide some partial results
towards a full characterization of the weights for which localization on 5
sites occurs with positive probability, and make some conjecture concerning the
almost sure behavior.Comment: 13 page
Excited Brownian motions as limits of excited random walks
We obtain the convergence in law of a sequence of excited (also called
cookies) random walks toward an excited Brownian motion. This last process is a
continuous semi-martingale whose drift is a function, say , of its local
time. It was introduced by Norris, Rogers and Williams as a simplified version
of Brownian polymers, and then recently further studied by the authors. To get
our results we need to renormalize together the sequence of cookies, the time
and the space in a convenient way. The proof follows a general approach already
taken by T\'oth and his coauthors in multiple occasions, which goes through
Ray-Knight type results. Namely we first prove, when is bounded and
Lipschitz, that the convergence holds at the level of the local time processes.
This is done via a careful study of the transition kernel of an auxiliary
Markov chain which describes the local time at a given level. Then we prove a
tightness result and deduce the convergence at the level of the full processes.Comment: v.3: main result improved: hyothesis of recurrence removed. To appear
in P.T.R.
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