标题： 随机Motzkin路径的极限与KPZ相关的渐近行为 显示英文标题

标题： Limits of Random Motzkin paths with KPZ related asymptotics

摘要： 我们研究长度为$L$的带有边和端点一般权重的 Motzkin 路径。 我们考察随机 Motzkin 路径的初始段和最终段的极限行为，将其视为从两个端点各自开始的一对过程，当$L$变得很大时。 然后我们研究所得过程的宏观极限，在两种不同的情形下，我们得到了在半直线上的 KPZ 方程的平稳测度描述中出现的马尔可夫过程，以及假设的半直线上的 KPZ 固定点的猜想平稳测度中出现的马尔可夫过程。 我们的结果依赖于 Al-Salam--Chihara 多项式在其正交区间上端附近的性质，以及$q$-Pochhammer 函数和$q$-Gamma 函数在$q\nearrow 1$时的极限性质。 显示英文摘要

摘要： We study Motzkin paths of length $L$ with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as $L$ becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half line. Our results rely on the behavior of the Al-Salam--Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the $q$-Pochhammer and $q$-Gamma functions as $q\nearrow 1$.