论文标题
半空间固定kardar-parisi-zhang方程
Half-space stationary Kardar-Parisi-Zhang equation
论文作者
论文摘要
我们研究了kardar-parisi-zhang（KPZ）方程的解决方案，以在正半线上的高度$ h（x，t）$的接口的随机生长，等效于连续体的自由能在半空间中，墙壁上有$ x = 0 $。边界条件$ \ partial_x h（x，t）| _ {x = 0} = a $对应于$ a <0 $的有吸引力的墙，并导致聚合物与临界值低于临界值$ a = -1/2 $的壁的绑定。在这里，我们选择初始条件$ h（x，0）$作为$ x> 0 $的布朗运动，带有漂移$ - （b+1/2）$。当$ a+b \ to -1 $时，解决方案是固定的，即$ h（\ cdot，t）$始终保持一项布朗尼运动，其漂移相同，直至全球高度偏移$ h（0，t）$。我们表明，在参数$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $的分布之间是不变的。对于任何$ a，b> -1/2 $，我们提供了一个精确的公式，以$ t $ t $的分布为特征，使用两种方法：bethe bethe ansatz和一种称为log -gamma聚合物的剥落，为此获得了矩公式。我们为各种参数$ a，b $分析了其大的时间渐近学。特别是，当$（a，b）\至（-1/2，-1/2）$时，关键的固定案例时，界面的波动受到通用分布的控制，类似于Baik-Rains分布在全线上的固定增长中产生的。它可以用简单的弗雷德尔姆决定因素来表达，也可以按照painlevéII的超越者表示。这为KPZ方程提供了一个类似物，这是Betea-Ferrari-Occelli最近在固定半空间最后一个passage-percolation的背景下获得的一些结果。从普遍性中，我们期望在两个模型中发现的限制分布都可以显示为重合。
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 $x=0$. The boundary condition $\partial_x h(x,t)|_{x=0}=A$ corresponds to an attractive wall for $A<0$, and leads to the binding of the polymer to the wall below the critical value $A=-1/2$. Here we choose the initial condition $h(x,0)$ to be a Brownian motion in $x>0$ with drift $-(B+1/2)$. When $A+B \to -1$, the solution is stationary, i.e. $h(\cdot,t)$ 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 $A,B > - 1/2$, 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 $(A, B) \to (-1/2, -1/2)$, 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.
