论文标题
部分可观测时空混沌系统的无模型预测
Sharp upper tail behavior of line ensembles via the tangent method
论文作者
论文摘要
我们开发了一种新的概率和几何方法，以获得与无限连续曲线的无限集合（也称为线集合物）的无限集合的上尾巴行为有关的几个尖锐的结果，也称为线结合体，满足了一些自然假设。这些论点使得对此类吉布斯措施的布朗重新采样不变性特性至关重要。我们获得了尖锐的单点上尾估计，表明零以大于$θ$的概率为$ \ exp（ - \ frac {4} {3}θ^θ^{3/2} {3/2}（1+o（1+o（1）））$。关键的中间步骤是在等于$θ$的零以零的情况下对配置文件进行精确理解。我们的方法还允许人们获得以前方法无法触及的多点渐近学。例如，我们证明了尖锐的两点上尾估计。我们的所有假设都显示出适用于通风$ _2 $过程的零温度案例，因此由于其与随机矩阵理论的连接以及新的两点渐近分析学的连接，为已经知道的一个点估计提供了新的证据。为了展示该方法的范围，我们仅在Gibbs措施类别的平稳性和极端性的假设下，在纯粹不可集成的设置中获得相同的结果。最后，除了相关性不平等外，所有假设也符合KPZ方程，这仍然是一个有趣的开放问题。我们的方法与Colomo-Sportiello引入的切线方法相似，而Aggarwal在六个Vertex模型的背景下实现了数学上实现的数学实现。
We develop a new probabilistic and geometric method to obtain several sharp results pertaining to the upper tail behavior of continuum Gibbs measures on infinite ensembles of random continuous curves, also known as line ensembles, satisfying some natural assumptions. The arguments make crucial use of Brownian resampling invariance properties of such Gibbs measures. We obtain sharp one-point upper tail estimates showing that the probability of the value at zero being larger than $θ$ is $\exp(-\frac{4}{3}θ^{3/2}(1+o(1)))$. A key intermediate step is developing a precise understanding of the profile when conditioned on the value at zero equaling $θ$. Our method further allows one to obtain multi-point asymptotics which were out of reach of previous approaches. As an example, we prove sharp explicit two-point upper tail estimates. All of our assumptions are shown to hold for the zero-temperature case of the Airy$_2$ process, thus yielding new proofs for one-point estimates already known due to its connections to random matrix theory, as well as new two-point asymptotics. To showcase the reach of the method, we obtain the same results in a purely non-integrable setting under only assumptions of stationarity and extremality in the class of Gibbs measures. Finally, all the assumptions are also shown to hold for the KPZ equation, except for a correlation inequality which remains an interesting open question. Our method bears resemblance to the tangent method introduced by Colomo-Sportiello and mathematically realized by Aggarwal in the context of the six-vertex model.
