8,672 research outputs found
Deterministic Random Walks on Regular Trees
Jim Propp's rotor router model is a deterministic analogue of a random walk
on a graph. Instead of distributing chips randomly, each vertex serves its
neighbors in a fixed order.
Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable
similarity of both models. If an (almost) arbitrary population of chips is
placed on the vertices of a grid and does a simultaneous walk in the
Propp model, then at all times and on each vertex, the number of chips on this
vertex deviates from the expected number the random walk would have gotten
there by at most a constant. This constant is independent of the starting
configuration and the order in which each vertex serves its neighbors.
This result raises the question if all graphs do have this property. With
quite some effort, we are now able to answer this question negatively. For the
graph being an infinite -ary tree (), we show that for any
deviation there is an initial configuration of chips such that after
running the Propp model for a certain time there is a vertex with at least
more chips than expected in the random walk model. However, to achieve a
deviation of it is necessary that at least vertices
contribute by being occupied by a number of chips not divisible by at a
certain time.Comment: 15 pages, to appear in Random Structures and Algorithm
Rotor walks on general trees
The rotor walk on a graph is a deterministic analogue of random walk. Each
vertex is equipped with a rotor, which routes the walker to the neighbouring
vertices in a fixed cyclic order on successive visits. We consider rotor walk
on an infinite rooted tree, restarted from the root after each escape to
infinity. We prove that the limiting proportion of escapes to infinity equals
the escape probability for random walk, provided only finitely many rotors send
the walker initially towards the root. For i.i.d. random initial rotor
directions on a regular tree, the limiting proportion of escapes is either zero
or the random walk escape probability, and undergoes a discontinuous phase
transition between the two as the distribution is varied. In the critical case
there are no escapes, but the walker's maximum distance from the root grows
doubly exponentially with the number of visits to the root. We also prove that
there exist trees of bounded degree for which the proportion of escapes
eventually exceeds the escape probability by arbitrarily large o(1) functions.
No larger discrepancy is possible, while for regular trees the discrepancy is
at most logarithmic.Comment: 32 page
On the speed of once-reinforced biased random walk on trees
We study the asymptotic behaviour of once-reinforced biased random walk
(ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk
has a bias towards or away from the root. We prove that in the setting of
multiplicative once-reinforcement the ORbRW can be recurrent even when the
underlying biased random walk is ballistic. We also prove that, on
Galton-Watson trees without leaves, the speed is positive in the transient
regime. Finally, we prove that, on regular trees, the speed of the ORbRW is
monotone decreasing in the reinforcement parameter when the underlying random
walk has high speed, and the reinforcement parameter is small
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