论文标题
log-gamma聚合物和KPZ方程的固定度量的半空间
Stationary measures for the log-gamma polymer and KPZ equation in half-space
论文作者
论文摘要
我们在半空间中构建了针对Kardar-Parisi-Zhang方程的固定度量的显式单参数家族，其原点为Neumann边界条件，以及在半空间中的Log-Gamma聚合物模型。固定度量是取决于边界条件的随机过程，以及与无穷大的漂移相关的参数。它们是根据布朗运动和伽玛随机步行的指数功能表示的。我们推测这些构成了这些模型的所有极端固定措施。通过与半空间Whittaker过程相关的对称参数证明了log-gamma聚合物结果，我们期望，该过程可能适用于其他可集成模型。 KPZ的结果是log-gamma聚合物结果的中间障碍极限，并确认了对这些固定措施的猜想描述：2105.15178。为了证明中间疾病的极限，我们提供了一个一般的半空间聚合物收敛框架，该框架扩展了Arxiv的作品：1804.09815，ARXIV：1901.09449，ARXIV：1202.4398。
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.
