We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the Kardar-Parisi-Zhang (KPZ) equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of Wu [Tightness and local fluctuation estimates for the KPZ line ensemble, 2021], Dimitrov and Matetski [Ann. Probab. 49 (2021), pp. 2477–2529], the KPZ line ensemble converges to the Airy line ensemble.For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with | h ( x ) | ≤ C ( 1 + | x | ) |h(x)|\le C(1+\sqrt {|x|}) for some C > 0 C>0 , or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts.The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates.

中文翻译：

---

排除过程和 KPZ 方程到 KPZ 不动点的收敛

我们表明，在 1:2:3 比例下，批判性地探测大空间和时间，有限范围非对称排除过程的高度函数和 Kardar-Parisi-Zhang (KPZ) 方程收敛到 KPZ 不动点，之前构造为通过精确公式限制完全不对称的简单排除过程。因此，基于 Wu [2021 年 KPZ 线系的紧密度和局部波动估计]、Dimitrov 和 Matetski [Ann. 概率。49 (2021), pp. 2477–2529]，KPZ 线系集合收敛于艾里线系。对于 KPZ 方程，我们能够从连续函数加上窄楔的有限集合开始。对于最近邻排除，我们可以用 | 对连续函数进行（离散化）h ( x ) | ≤ C ( 1 + | x | ) |h(x)|\le C(1+\sqrt {|x|}) 对于一些 C > 0 C>0 或一个窄楔形。对于非最近邻排除，我们目前仅限于一类（随机）初始数据，在紧凑的均匀收敛拓扑中的连续函数中密集。该方法是通过比较有限范围排除的转移概率过程和使用能量估计的完全不对称的简单排除过程。