论文标题
三速弹道式灭绝的组合普遍性
Combinatorial universality in three-speed ballistic annihilation
论文作者
论文摘要
我们考虑一个一维颗粒系统，以$ \ { - 1,0，+1 \} $在$ \ { - 1,0，+1 \} $上的对称分布独立选择的恒定速度上移动，并在碰撞时歼灭 - 与三重碰撞，在两个移动粒子中均匀随机选择幸存者。当系统包含无限的许多粒子（其起始位置是通过更新过程给出的）时，由于静态粒子的密度越过$ 1/4 $，因此证明了相变发生的相变（见Arxiv：1811.08709）。值得注意的是，观察到这种临界价值以及某些其他统计数据不依赖于互相的分布。在本文中，我们通过证明对具有固定但随机洗牌的有限粒子系统的有限粒子系统的更强有力的陈述来进一步研究这一普遍性。我们给出了两个证据，一个通过归纳允许进行明确的计算，另一个通过更直接的比较。该结果需要一种新的非平凡独立性，特别是在无限模型中给定时间访问存活的静态颗粒的密度。最后，在不对称的情况下，事实证明，进一步的类似独立性可以保持持有，包括与一般行为形成鲜明对比的伽马分布式互相的惊人属性。
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.
