In this paper we consider the Higher Spin Six Vertex Model on the lattice $${\mathbb {Z}}_{\ge 2} \times {\mathbb {Z}}_{\ge 1}$$ . We first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions. Exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space-time limit for the stationary Higher Spin Six Vertex Model. In particular, along the characteristic line we recover Baik–Rains fluctuations with size of characteristic exponent 1/3. We also consider some of the main degenerations of the Higher Spin Six Vertex Model and we adapt our analysis to the relevant cases of the q-Hahn particle process and of the Exponential Jump Model.

中文翻译：

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平稳随机高自旋六顶点模型和 q-Whittaker 测度

在本文中，我们考虑格子上的高自旋六顶点模型 $${\mathbb {Z}}_{\ge 2} \times {\mathbb {Z}}_{\ge 1}$$ 。我们首先确定了一系列平移不变测度，然后我们研究了具有某些随机边界条件的模型的高度函数的单点分布。我们获得的精确公式被证明是有用的，以建立稳态高自旋六顶点模型在长时空限制内高度分布的渐近线。特别是，沿着特征线，我们恢复了特征指数大小为 1/3 的 Baik-Rains 波动。我们还考虑了高自旋六顶点模型的一些主要退化，并将我们的分析调整到 q-Hahn 粒子过程和指数跳跃模型的相关案例。