44 research outputs found
Multi-excited random walks on integers
We introduce a class of nearest-neighbor integer random walks in random and
non-random media, which includes excited random walks considered in the
literature. At each site the random walker has a drift to the right, the
strength of which depends on the environment at that site and on how often the
walker has visited that site before. We give exact criteria for recurrence and
transience and consider the speed of the walk.Comment: 25 pages, 3 figure
The zero-one law for planar random walks in i.i.d. random environments revisited
In this note we present a simplified proof of the zero-one law by Merkl and
Zerner (2001) for directional transience of random walks in i.i.d. random
environments (RWRE) on the square lattice. Also, we indicate how to construct a
two-dimensional counterexample in a non-uniformly elliptic and stationary
environment which has better ergodic properties than the example given by Merkl
and Zerner.Comment: 9 page
Shortest spanning trees and a counterexample for random walks in random environments
We construct forests that span , , that are stationary
and directed, and whose trees are infinite, but for which the subtrees attached
to each vertex are as short as possible. For , two independent copies
of such forests, pointing in opposite directions, can be pruned so as to become
disjoint. From this, we construct in a stationary, polynomially mixing
and uniformly elliptic environment of nearest-neighbor transition probabilities
on , for which the corresponding random walk disobeys a certain
zero--one law for directional transience.Comment: Published at http://dx.doi.org/10.1214/009117905000000783 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Lyapunov exponents of Green's functions for random potentials tending to zero
We consider quenched and annealed Lyapunov exponents for the Green's function
of , where the potentials , are i.i.d.
nonnegative random variables and is a scalar. We present a
probabilistic proof that both Lyapunov exponents scale like as
tends to 0. Here the constant is the same for the quenched as for
the annealed exponent and is computed explicitly. This improves results
obtained previously by Wei-Min Wang. We also consider other ways to send the
potential to zero than multiplying it by a small number.Comment: 16 pages, 3 figures. 1 figure added, very minor corrections. To
appear in Probability Theory and Related Fields. The final publication is
available at http://www.springerlink.com, see
http://www.springerlink.com/content/p0873kv68315847x/?p=4106c52fc57743eba322052bb931e8ac&pi=21