三速弹道湮灭中的组合普适性
Combinatorial universality in three-speed ballistic annihilation
摘要：
我们考虑一个一维系统的粒子，以恒定的速度移动独立选择根据$\{-1,0,+1\}$上的对称分布，并在碰撞时湮灭，在三重碰撞的情况下，一个均匀随机选择的幸存者之间的两个移动粒子。当系统包含无穷多个粒子时，其起始位置由更新过程给出，当静态粒子的密度超过值$1/4$时，相变被证明发生（参见arXiv：1811.08709）。值得注意的是，这个临界值，以及某些其他统计，被观察到不依赖于间距的分布。在本文中，我们进一步调查这种普遍性，证明了一个更强的声明有限系统的粒子固定，但随机洗牌，间距。我们给出了两个证明，一个是由一个感应允许明确的计算，一个是由一个更直接的比较。这个结果需要一个新的非平凡的独立属性，特别是在给定的时间在无限模型中的幸存的静态粒子的密度。最后，在不对称的情况下，进一步类似的独立性被证明保持持有，包括一个引人注目的伽玛分布的间距，与一般的行为形成对比的属性。
英文摘要：
We consider a one-dimensional system of particles, moving at constant velocities chosen independently according to a symmetric distribution on $\{-1,0,+1\}$, and annihilating upon collision -- with, in case of triple collision, a uniformly random choice of survivor among the two moving particles. When the system contains infinitely many particles, whose starting locations are given by a renewal process, a phase transition was proved to happen (see arXiv:1811.08709) as the density of static particles crosses the value $1/4$. Remarkably, this critical value, along with certain other statistics, was observed not to depend on the distribution of interdistances. In the present paper, we investigate further this universality by proving a stronger statement about a finite system of particles with fixed, but randomly shuffled, interdistances. We give two proofs, one by an induction allowing explicit computations, and one by a more direct comparison. This result entails a new nontrivial independence property that in particular gives access to the density of surviving static particles at a given time in the infinite model. Finally, in the asymmetric case, further similar independence properties are proved to keep holding, including a striking property of gamma distributed interdistances that contrasts with the general behavior.
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