We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document}x=0. The boundary condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x h(x,t)|_{x=0}=A$$\end{document}∂xh(x,t)|x=0=A corresponds to an attractive wall for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A<0$$\end{document}A<0, and leads to the binding of the polymer to the wall below the critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=-1/2$$\end{document}A=-1/2. Here we choose the initial condition h(x, 0) to be a Brownian motion in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>0$$\end{document}x>0 with drift \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(B+1/2)$$\end{document}-(B+1/2). When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A+B \rightarrow -1$$\end{document}A+B→-1, the solution is stationary, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\cdot ,t)$$\end{document}h(·,t) remains at all times a Brownian motion with the same drift, up to a global height shift h(0, t). We show that the distribution of this height shift is invariant under the exchange of parameters A and B. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A,B > - 1/2$$\end{document}A,B>-1/2, we provide an exact formula characterizing the distribution of h(0, t) at any time t, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A, B. In particular, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, B) \rightarrow (-1/2, -1/2)$$\end{document}(A,B)→(-1/2,-1/2), the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.

中文翻译：

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半空间静止 Kardar-Parisi-Zhang 方程

我们研究了高度为 h(x, t) 的界面在正半线上的随机增长的 Kardar-Parisi-Zhang (KPZ) 方程的解，等效于半空间中连续介质定向聚合物的自由能\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\奇边距}{-69pt} \begin{document}$$x=0$$\end{document}x=0。边界条件 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\奇边距}{-69pt} \begin{document}$$\partial _x h(x,t)|_{x=0}=A$$\end{document}∂xh(x, t)|x=0=A 对应于 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{ 的有吸引力的墙mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A<0$$\end{document}A<0，并导致聚合物与下面的墙壁结合临界值 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\奇边距}{-69pt} \begin{document}$$A=-1/2$$\end{document}A=-1/2。这里我们选择初始条件 h(x, 0) 在 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek 中成为布朗运动} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>0$$\end{document}x>0 withdrift \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym } \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(B+1/2 )$$\end{文档}-(B+1/2)。当 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin} {-69pt} \begin{document}$$A+B \rightarrow -1$$\end{document}A+B→-1, 解决方案是固定的，即 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \ setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\cdot ,t)$$\end{document}h(·,t) 始终保持具有相同漂移的布朗运动，直到全局高度偏移 h(0, t)。我们证明了这个高度偏移的分布在参数 A 和 B 的交换下是不变的。对于任何 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A,B > - 1/2$$\end{document}A,B >-1/2，我们提供了一个精确的公式来表征 h(0, t) 在任何时间 t，使用两种方法：复制 Bethe ansatz 和称为 log-gamma 聚合物的离散化，获得了矩公式。我们针对各种参数范围 A、B 分析其大时间渐近线。特别是，当 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{ amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, B) \rightarrow (-1/2, -1/2)$$\ end{document}(A,B)→(-1/2,-1/2)，临界平稳情况，界面的波动受一个类似于 Baik-Rains 分布的普遍分布控制全线。它可以用一个简单的 Fredholm 行列式来表达，或者等价地用 Painlevé II 超越式来表达。这为 Betea-Ferrari-Occelli 最近在静止半空间最后通道渗透的背景下获得的一些结果提供了 KPZ 方程的类比。从普遍性来看，我们期望在两个模型中发现的限制分布可以证明是一致的。