Collision probabilities in the rarefaction fan of asymmetric exclusion processes
Pablo A. Ferrari, Patricia Goncalves, James B. Martin
备注：
17 pages. This version is substantially rewritten. In particular an error in the statement of one of the results is corrected, the notation is changed, and new results on coexistence probabilities in a multi-type version of the corner growth model are added
We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate $p\in(1/2,1]$ and to the left at rate $1-p$, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is $(1+p)/3p$, where a_collision_ occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called "basic coupling", eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case ($p=1$) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.
我们确定了弹道和扩散重标度的极限分布的第二类粒子的位置在广泛的非对称和对称的相互作用粒子系统建立流体动力学行为，分别（包括零范围，厌世和许多其他模型）。初始条件是一个步骤的配置文件，在一些经典的情况下，不对称的模式，产生一个稀疏风扇的情况。我们还指出了一个模型与非凹，非凸的流体动力学，重新标度的第二类粒子分布有连续和离散的副本。这些结果来自于P. A. Ferrari和C. Kipnis的论点（“Second class particles in the rarefaction fan”，Ann.Inst.H.2004）。Poincaré，31，1995）的完全不对称的简单排它过程。主要的新颖性是引入了一个有符号的耦合措施作为初始数据，但结果在一个适当的概率初始分布的网站的第二类粒子，使扩展成为可能。我们还揭示了在充分的一般性一个非常有趣的不变性的一个网站的边缘分布的过程下的第二类粒子，特别是证明了我们的选择的内在性的初始分布。最后给出了第二类粒子-反粒子对生存概率的一个较低的估计。
