Correction To: Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension
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Correction To: Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension Eric Foxall1 · Hanbaek Lyu2 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Correction To: J Stat Phys https://doi.org/10.1007/s10955-018-2004-2 1 Summary There is an error in the proof of Lemma 4.1. Specifically, there is an ambiguity in the definition of Z t (x, y), which is such that, depending on how one interprets the ambiguity, the probabilistic interpretation of either E[ y∈Z Z t (0, y)] or E[ y∈Z Z t (y, 0)] given afterward is false. Instead of trying to prove the claim of Lemma 4.1—that p(t) ≥ q(t) for all t ≥ 0—which I suspect may be false, we prove the similar, but asymptotic statement lim supt→∞ q(t) ≤ inf t≥0 p(t). We then show that this statement can be used in place of p(t) ≥ q(t), which is not hard to do. This is done in Section 4 below.
2 Missing Concept: Particle Identity It appears to me that the reason for which the error was not caught is the absence of a precise notion of particle identity, that is, a criterion that determines whether a particle that exists at time t is the same particle, or a different particle, from one that exists at a time s = t. As described in Sect. 3 of the paper, edge particles change over time either by movement, or by collisions of various types. It should already be clear how to track a particle over time when it does not move, or when it moves but does not collide, but it is perhaps not immediately clear whether a particle that emerges from a collision is “the same particle” as one of the particles that took part in the collision. For both κ = 3 and κ = 4, collisions always involve two particles, and always result in either 0 or 1 particles. Moreover, in every collision there is
The original article can be found online at https://doi.org/10.1007/s10955-018-2004-2.
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Hanbaek Lyu [email protected] Eric Foxall [email protected]
1
Department of Mathematics, University of Alberta, Edmonton, AB, Canada
2
Department of Mathematics, The Ohio State University, Columbus, OH, USA
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E. Foxall, H. Lyu
a directed particle (the “attacking particle”) that moves onto a space (the “collision site”) that is already occupied by another particle (the “attacked particle”). Moreover, in cases where a particle remains after the collision, it is always located at the collision site. It seems that the most natural definition is to declare that the remaining particle, when one exists, is the same particle as the attacking particle – of course, in the event of annihilation, both particles are destroyed. In this way, whenever a collision does not result in annihilation of both particles, the attacking particle survives, while the attacked particle is destroyed. This definition ensures that particles can never be created, only destroyed. In addition, the attacking particle may change type. It is also worth noting that since a blockade does not move, it cannot be the attacking particle in a collision, so can never change type – it can either remain where
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