A rotor configuration in Z^d where Schramm's bound of escape rates attains

TL;DR

This paper investigates rotor walks in Z^d, demonstrating that under certain initial configurations, the escape rates of particles reach known upper bounds, with distinct behaviors in dimensions 2 and 3 or higher.

Contribution

It constructs specific initial rotor configurations in Z^d that achieve the maximal escape rates, extending understanding of rotor-router dynamics and bounds in different dimensions.

Findings

Escape rate in d>=3 attains Schramm's upper bound.

In d=2, escape particles are of order n/log(n) with a limit matching pi/2.

Methods from rotor-router aggregation are employed for analysis.

Abstract

Rotor walk is deterministic counterpart of random walk on graphs. We study that under a certain initial configuration in Z^d, n particles perform rotor walks from the origin consecutively. They would stop if they hit the origin or infinity. When the dimension d>=3, the escape rate exists and it attains the upper bound of O. Schramm. When the dimension d=2, the numbers of the particles escaping to infinity are of order n/log(n). The limit of their quotient exist and also attains the upper bound of L.Florescu,S.Ganguly,L.Levine,Y.Peres which equals to frac{pi}{2}. We use the results and the methods of the outer estimate for rotor-router aggregation in L.Levine and Y.Peres' previous paper.