论文标题
随机六个vertex模型与通风过程的两点收敛
Two-point convergence of the stochastic six-vertex model to the Airy process
论文作者
论文摘要
在本文中，我们考虑象限中随机的六个vertex模型始于步骤初始数据。经过长时间的$ t $，众所周知，单点高度功能波动是$ t^{1/3} $的顺序，并由tracy-widom分布管理。我们证明，高度函数的两点分布，由$ t^{2/3} $水平重新缩放，并通过$ t^{1/3} $垂直分配，收敛到通风过程的两点分布。该结果的起点是由硼丁蛋白 - 布菲托夫轮毂在随机的六个vertex模型和上升的霍尔 - 小木过程（平面分区的一定量度）之间发现的最新连接。使用MacDonald差异运算符，我们为升高的Hall-Littlewood工艺获得了两点可观察物的公式，该过程对于六个vertex模型，它可以访问其高度函数的关节累积分布函数。对这些可观察结果的仔细渐近分析给出了对模型参数的一定限制，从而给出了两点收敛的结果。
In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of order $T^{1/3}$ and governed by the Tracy-Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $T^{2/3}$ and vertically by $T^{1/3}$, converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin-Bufetov-Wheeler between the stochastic six-vertex model and the ascending Hall-Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall-Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.
