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arXiv · 1405.3400

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

Daiwei He
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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.

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Daiwei He. 2014-05-14. A rotor configuration in Z^d where Schramm's bound of escape rates attains. https://arxiv.org/abs/1405.3400

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