# Spherical asymptotics for the rotor-router model in Zd

@article{Levine2005SphericalAF, title={Spherical asymptotics for the rotor-router model in Zd}, author={Lionel Levine and Yuval Peres}, journal={Indiana University Mathematics Journal}, year={2005}, volume={57}, pages={431-450} }

The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region in Z d , and use this to prove that the shape of the rotor-router aggregation model in Z d , suitably rescaled, converges to a Euclidean ball in R d .

## 48 Citations

### Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

- Mathematics
- 2008

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is…

### Rotor-Router Aggregation on the Layered Square Lattice

- MathematicsElectron. J. Comb.
- 2010

It is shown that for a certain choice of initial rotors, the occupied cluster of occupied sites grows as a perfect diamond.

### A loop reversibility and subdiffusion of the rotor-router walk

- Mathematics
- 2015

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…

### Chip-Firing and Rotor-Routing on Z d and on Trees

- Computer Science
- 2008

In the regular tree, the sandpile group is used to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball, which is close to circular.

### Deterministic Random Walks for Rapidly Mixing Chains

- MathematicsSIAM J. Discret. Math.
- 2018

The discrepancy of the number of tokens at a single vertex between the functional-router model and its corresponding Markov chain is investigated, and an upper bound in terms of the mixing time of the MarkovChain is given.

### Spiral structures in the rotor-router walk

- Mathematics
- 2015

We study the rotor-router walk on the infinite square lattice with the outgoing edges at each lattice site ordered clockwise. In the previous paper (Papoyan et al 2015 J. Phys. A: Math. Theor. 48…

### Limiting shapes for deterministic internal growth models

- Mathematics
- 2007

We study the rotor router model and two deterministic sandpile models. For the rotor router model in Z d , Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the…

### Deterministic Random Walks

- Mathematics, Computer ScienceANALCO
- 2006

Jim Propp's P-machine, also known as 'rotor router model' is a simple deterministic process that simulates a random walk on a graph, which shows that, independent of the starting configuration, the number of chips on this vertex deviates from the expected number of Chips in the random walk model by at most a constant c1, which is approximately 2.29.

### Limiting Shapes for Deterministic Centrally Seeded Growth Models

- Mathematics
- 2007

Abstract
We study the rotor router model and two deterministic sandpile models. For the rotor router model in ℤd, Levine and Peres proved that the limiting shape of the growth cluster is a sphere.…

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