Stationary measures for the log-gamma polymer and KPZ equation in half-space
Stationary measures for the log-gamma polymer and KPZ equation in half-space
复制标题
DOI：
10.1214/23-aop1634
复制
发表时间：
2022-03
期刊：
影响因子：
--
通讯作者：
Guillaume Barraquand;Ivan Corwin
中科院分区：
文献类型：
--
作者：
Guillaume Barraquand;Ivan Corwin
我们构造显式单参数家庭的Kardar-Parisi-Zhang方程在半空间与Neumann边界条件的起源，以及在半空间的对数伽玛聚合物模型的平稳措施。平稳测度是随机过程，它依赖于边界条件以及与无穷远处漂移有关的参数。它们用布朗运动和伽玛随机游动的指数函数表示。我们猜想，这些构成所有极端平稳措施，这些模型。通过与半空间Whittaker过程相关的对称性论证证明了对数-γ聚合物的结果，我们期望这一结果可以应用于其他可积模型。KPZ结果作为log-γ聚合物结果的中间无序限，并证实了arXiv：2105.15178中这些固定测量的结构描述。为了证明中间无序极限，我们提供了一个一般的半空间聚合物收敛框架，扩展了arXiv：1804.09815，arXiv：1901.09449，arXiv：1202.4398的工作。
We construct explicit one-parameter families of stationary measures for the Kardar-Parisi-Zhang equation in half-space with Neumann boundary conditions at the origin, as well as for the log-gamma polymer model in a half-space. The stationary measures are stochastic processes that depend on the boundary condition as well as a parameter related to the drift at infinity. They are expressed in terms of exponential functions of Brownian motions and gamma random walks. We conjecture that these constitute all extremal stationary measures for these models. The log-gamma polymer result is proved through a symmetry argument related to half-space Whittaker processes which we expect may be applicable to other integrable models. The KPZ result comes as an intermediate disorder limit of the log-gamma polymer result and confirms the conjectural description of these stationary measures from arXiv:2105.15178. To prove the intermediate disorder limit, we provide a general half-space polymer convergence framework that extends works of arXiv:1804.09815, arXiv:1901.09449, arXiv:1202.4398.