论文标题
对称排除过程中的牙胶法律
Gumbel laws in the symmetric exclusion process
论文作者
论文摘要
我们考虑从无限粒子步长配置开始的$ \ mathbb {z} $上的对称排除粒子系统，其中最大值的右侧没有粒子。我们表明，缩放位置$ x_t/（σb_t） - 时间$ t $在时间$ t $上的a_t $收敛到牙胶极限定律，其中$ b_t = \ sqrt {t/\ log t} $，$ a_t = \ log log（t/（t/（t/（t/（t/is）的随机步行）概率。这项工作解决了Arratia（1983）中留下的问题。 此外，为了研究领先粒子背后的质量质量的影响，我们考虑了由$ l $粒子块组成的初始概况，其中$ l \ to \ infty $ as $ t \ to \ to \ infty $。当$ l $以$ t $偏差时，在适当缩放下的牙龈限制法律将以$ x_t $的价格获得。特别是，当$ l $是订单$ b_t $的转换，高于$ x_t $的位移与在无限粒子步骤配置文件下的位移相似，而在其下方则是$ \ sqrt {t \ log l} $的位移。 证明是基于最近开发的对称排除系统的负依赖性特性。还对从不对称最近邻居排除的步骤曲线开始的最正确粒子的行为进行了备注，这补充了已知的结果。
We consider the symmetric exclusion particle system on $\mathbb{Z}$ starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position $X_t/(σb_t) - a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law, where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2π}\log t))$, and $σ$ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of $L$ particles, where $L \to \infty$ as $t \to \infty$. Gumbel limit laws, under appropriate scaling, are obtained for $X_t$ when $L$ diverges in $t$. In particular, there is a transition when $L$ is of order $b_t$, above which the displacement of $X_t$ is similar to that under a infinite particle step profile, and below which it is of order $\sqrt{t\log L}$. Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.
