1,307 research outputs found
A law of large numbers for the range of rotor walks on periodic trees
The aim of the current work is to prove a law of large numbers for the range
size of recurrent rotor walks with random initial configuration on a general
class of trees, called periodic trees or directed covers of graphs.Comment: New references added; 21 pages, 1 figure. Appendix on rotor-recurrent
trees that contain rotor-transient subtrees. arXiv admin note: text overlap
with arXiv:1203.147
Range and speed of rotor walks on trees
We prove a law of large numbers for the range of rotor walks with random
initial configuration on regular trees and on Galton-Watson trees. More
precisely, we show that on the classes of trees under consideration, even in
the case when the rotor walk is recurrent, the range grows at linear speed. We
also show the existence of the speed for such rotor walks.Comment: Final version; accepted in Journal of Theoretical Probability 201
Random walks with local memory
We prove a quenched invariance principle for a class of random walks in
random environment on , where the walker alters its own
environment. The environment consists of an outgoing edge from each vertex. The
walker updates the edge at its current location to a new random edge
(whose law depends on ) and then steps to the other endpoint of . We
show that a native environment for these walks (i.e., an environment that is
stationary in time from the perspective of the walker) consists of the wired
uniform spanning forest oriented toward the walker, plus an independent
outgoing edge from the walker.Comment: 29 pages, 7 figure
Rotor walks on transient graphs and the wired spanning forest
We study rotor walks on transient graphs with initial rotor configuration
sampled from the oriented wired uniform spanning forest (OWUSF) measure. We
show that the expected number of visits to any vertex by the rotor walk is at
most equal to the expected number of visits by the simple random walk. In
particular, this implies that this walk is transient. When these two numbers
coincide, we show that the rotor configuration at the end of the process also
has the law of OWUSF. Furthermore, if the graph is vertex-transitive, we show
that the average number of visits by consecutive rotor walks converges to
the Green function of the simple random walk as tends to infinity. This
answers a question posed by Florescu, Ganguly, Levine, and Peres (2014).Comment: 32 pages, 3 figure
A rotor configuration with maximum escape rate
Rotor walk is a deterministic analogue of simple random walk. For any given
graph, we construct a rotor configuration for which the escape rate of the
corresponding rotor walk is equal to the escape rate of simple random walk, and
thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).Comment: 6 page
Internal Aggregation Models on Comb Lattices
The two-dimensional comb lattice is a natural spanning tree of the
Euclidean lattice . We study three related cluster growth models
on : internal diffusion limited aggregation (IDLA), in which random
walkers move on the vertices of until reaching an unoccupied site where
they stop; rotor-router aggregation in which particles perform deterministic
walks, and stop when reaching a site previously unoccupied; and the divisible
sandpile model where at each vertex there is a pile of sand, for which, at each
step, the mass exceeding 1 is distributed equally among the neighbours. We
describe the shape of the divisible sandpile cluster on , which is then
used to give inner bounds for IDLA and rotor-router aggregation.Comment: 23 pages, 4 figure
Euler tours and unicycles in the rotor-router model
A recurrent state of the rotor-routing process on a finite sink-free graph
can be represented by a unicycle that is a connected spanning subgraph
containing a unique directed cycle. We distinguish between short cycles of
length 2 called "dimers" and longer ones called "contours". Then the
rotor-router walk performing an Euler tour on the graph generates a sequence of
dimers and contours which exhibits both random and regular properties. Imposing
initial conditions randomly chosen from the uniform distribution we calculate
expected numbers of dimers and contours and correlation between them at two
successive moments of time in the sequence. On the other hand, we prove that
the excess of the number of contours over dimers is an invariant depending on
planarity of the subgraph but not on initial conditions. In addition, we
analyze the mean-square displacement of the rotor-router walker in the
recurrent state.Comment: 17 pages, 4 figures. J. Stat. Mech. (2014
A Loop Reversibility and Subdiffusion of the Rotor-Router Walk
The rotor-router model on a graph describes a discrete-time walk accompanied
by the deterministic evolution of configurations of rotors randomly placed on
vertices of the graph. We prove the following property: if at some moment of
time, the rotors form a closed clockwise contour on the planar graph, then the
clockwise rotations of rotors generate a walk which enters into the contour at
some vertex , performs a number of steps inside the contour so that the
contour formed by rotors becomes anti-clockwise, and then leaves the contour at
the same vertex . This property generalizes the previously proved theorem
for the case when the rotor configuration inside the contour is a cycle-rooted
spanning tree, and all rotors inside the contour perform a full rotation. We
use the proven property for an analysis of the sub-diffusive behavior of the
rotor-router walk.Comment: 15 pages, 5 figure
Discrete low-discrepancy sequences
Holroyd and Propp used Hall's marriage theorem to show that, given a
probability distribution pi on a finite set S, there exists an infinite
sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the
number of i in [1,k] with s_i = s differs from k pi(s) by at most 1. We prove a
generalization of this result using a simple explicit algorithm. A special case
of this algorithm yields an extension of Holroyd and Propp's result to the case
of discrete probability distributions on infinite sets.Comment: Since posting the preprint, we have learned that our main result was
proved by Tijdeman in the 1970s and that his proof is the same as our
Rotor-Router Aggregation on the Comb
We prove a shape theorem for rotor-router aggregation on the comb, for a
specific initial rotor configuration and clockwise rotor sequence for all
vertices. Furthermore, as an application of rotor-router walks, we describe the
harmonic measure of the rotor-router aggregate and related shapes, which is
useful in the study of other growth models on the comb. We also identify the
shape for which the harmonic measure is uniform. This gives the first known
example where the rotor-router cluster has non-uniform harmonic measure, and
grows with different speeds in different directions.Comment: 23 pages, 4 figure
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