A stochastic process called vertex-reinforced random walk (VRRW) is
defined in Pemantle [Ann. Probab. 16 1229–1241] . We consider
this process in the case where the underlying graph is an infinite chain (i.e.,
the one-dimensional integer lattice). We show that the range is almost surely
finite, that at least five points are visited infinitely often almost surely
and that with positive probability the range contains exactly five points.
There are always points visited infinitely often but at a set of times of zero
density, and we show that the number of visits to such a point to time $n$ may
be asymptotically $n ^[\alpha}$ for a dense set of values $\alpha \in (0,1)$.
The power law analysis relies on analysis of a related urn model.
Publié le : 1999-07-14
Classification:
Vertex-reinforced random walk,
Reinforced random walk,
VRRW,
Urn model,
Bernard Friedman’s urn,
60G17,
60J20
@article{1022677452,
author = {Pemantle, Robin and Volkov, Stanislav},
title = {Vertex-Reinforced Random Walk on Z Has Finite Range},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 1368-1388},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677452}
}
Pemantle, Robin; Volkov, Stanislav. Vertex-Reinforced Random Walk on Z Has Finite Range. Ann. Probab., Tome 27 (1999) no. 1, pp. 1368-1388. http://gdmltest.u-ga.fr/item/1022677452/