竞争界面的相变
A phase transition for competition interfaces
摘要：
我们研究了在一个与最后一次渗流相关联的生长模型中两个生长团簇之间的竞争界面。当初始未占用集近似为一个锥时，我们证明了这个接口具有概率为1的渐近方向。该方向的行为取决于圆锥的角度$θ$：对于$θ\geq180^{\circ}$，方向是确定性的，而对于$θ<180^{\circ}$，方向是随机的，并且在某些情况下可以明确给出其分布。我们还得到了界面在其渐近方向附近的涨落的部分结果。生长模型中竞争界面的演化可以映射到完全非对称简单排它过程中第二类粒子的路径上，从界面极限方向的存在性出发，得到了第二类粒子位置在大时间内的强大数律（可能有随机极限）的一个新的、相当自然的证明.
英文摘要：
We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle $θ$ of the cone: for $θ\geq180^{\circ}$, the direction is deterministic, while for $θ<180^{\circ}$, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.
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