We consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling-invariant Markov process defined in [MQR17] and believed to govern the long-time, large-scale fluctuations for all models in the KPZ universality class. Brownian last-passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.

中文翻译：

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单面反射布朗运动和 KPZ 不动点

我们考虑单边反射布朗运动系统，该系统与布朗最后通道渗透处于变分对偶状态。我们证明了它具有可积的转移概率，用 Hermite 多项式和指数随机游走的命中时间表示，并且它收敛于 KPZ 不动点的 1:2:3 缩放限制，定义的缩放不变马尔可夫过程[MQR17] 并被认为控制 KPZ 普适性类中所有模型的长期、大规模波动。最近在 [DOV18] 中显示了布朗最后通道渗流收敛到艾里片（或定向景观），在那里定义为艾里线系综的功能的强极限。这建立了根据艾里表的 KPZ 不动点的变分公式。