论文标题
随机六个顶点模型中KPZ状态的不可逆转的马尔可夫动力学和流体动力学
Irreversible Markov Dynamics and Hydrodynamics for KPZ States in the Stochastic Six Vertex Model
论文作者
论文摘要
我们在二维方晶格上定义的离散高度功能上介绍了马尔可夫增长过程。每个高度函数对应于无限方晶格上六个顶点模型的配置。我们专注于随机六个顶点模型，该模型对应于铁电状态内的特定两参数重量家族。据信（并部分证明，请参见Aggarwal，Arxiv：2004.13272），随机六个顶点模型显示出两种类型的非平凡的纯净（即翻译不变和赤牙）状态，即KPZ和液体。这些阶段具有非常不同的远程相关结构。我们构建的马尔可夫过程将在整个平面中保留KPZ纯状态。我们还表明，在圆环上放置的相同过程保留了通用六个顶点权重的任意吉布斯度量（不一定在铁电状态下）。 我们的动力学自然来自于六个顶点模型的杨巴克斯特方程，该方程是通过其bufetov-petrov（Arxiv：1712.04584）首次使用的技术。我们构建的动态是不可逆转的。特别是，高度函数具有非零的平均漂移。在每个kpz纯状态下，我们明确地计算出平均漂移（也称为电流）作为斜率的函数。我们使用它来分析作用于四分之一平面随机六个顶点配置的过程的非平稳版本的流体动力学。在四分之一平面模型中的固定时间极限形状是在硼丹 - 果蛋白 - 戈林（Arxiv：1407.6729）中获得的。
We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice. We focus on the stochastic six vertex model corresponding to a particular two-parameter family of weights within the ferroelectric regime. It is believed (and partially proven, see Aggarwal, arXiv:2004.13272) that the stochastic six vertex model displays nontrivial pure (i.e., translation invariant and ergodic) Gibbs states of two types, KPZ and liquid. These phases have very different long-range correlation structure. The Markov processes we construct preserve the KPZ pure states in the full plane. We also show that the same processes put on the torus preserve arbitrary Gibbs measures for generic six vertex weights (not necessarily in the ferroelectric regime). Our dynamics arise naturally from the Yang-Baxter equation for the six vertex model via its bijectivisation, a technique first used in Bufetov-Petrov (arXiv:1712.04584). The dynamics we construct are irreversible; in particular, the height function has a nonzero average drift. In each KPZ pure state, we explicitly compute the average drift (also known as the current) as a function of the slope. We use this to analyze the hydrodynamics of a non-stationary version of our process acting on quarter plane stochastic six vertex configurations. The fixed-time limit shapes in the quarter plane model were obtained in Borodin-Corwin-Gorin (arXiv:1407.6729).
