417 research outputs found
Maximal Displacement for Bridges of Random Walks in a Random Environment
It is well known that the distribution of simple random walks on
conditioned on returning to the origin after steps does not depend on , the probability of moving to the right. Moreover, conditioned on
the maximal displacement converges in
distribution when scaled by (diffusive scaling).
We consider the analogous problem for transient random walks in random
environments on . We show that under the quenched law
(conditioned on the environment ), the maximal displacement of the
random walk when conditioned to return to the origin at time is no longer
necessarily of the order . If the environment is nestling (both
positive and negative local drifts exist) then the maximal displacement
conditioned on returning to the origin at time is of order
, where the constant depends on the law on
environment. On the other hand, if the environment is marginally nestling or
non-nestling (only non-negative local drifts) then the maximal displacement
conditioned on returning to the origin at time is at least
and at most for any
.
As a consequence of our proofs, we obtain precise rates of decay for
. In particular, for certain non-nestling environments we
show that
with explicit constants .Comment: Revised version, 19 pages, 1 figure To appear in: AIHP Prob. & Sta
Routing on trees
We consider three different schemes for signal routing on a tree. The
vertices of the tree represent transceivers that can transmit and receive
signals, and are equipped with i.i.d. weights representing the strength of the
transceivers. The edges of the tree are also equipped with i.i.d. weights,
representing the costs for passing the edges. For each one of our schemes, we
derive sharp conditions on the distributions of the vertex weights and the edge
weights that determine when the root can transmit a signal over arbitrarily
large distances
Annealed deviations of random walk in random scenery
Let be a -dimensional {\it random walk in random
scenery}, i.e.,
with a random walk in
and an i.i.d. scenery, independent of the walk. The
walker's steps have mean zero and finite variance. We identify the speed and
the rate of the logarithmic decay of for various choices
of sequences in . Depending on and the upper
tails of the scenery, we identify different regimes for the speed of decay and
different variational formulas for the rate functions. In contrast to recent
work \cite{AC02} by A. Asselah and F. Castell, we consider sceneries {\it
unbounded} to infinity. It turns out that there are interesting connections to
large deviation properties of self-intersections of the walk, which have been
studied recently by X. Chen \cite{C03}.Comment: 32 pages, revise
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