Exceptional times when the KPZ fixed point violates Johansson’s conjecture on maximizer uniqueness
Exceptional times when the KPZ fixed point violates Johansson’s conjecture on maximizer uniqueness
复制标题
DOI：
10.1214/22-ejp898
复制
发表时间：
2021-01
影响因子：
1.4
通讯作者：
Ivan Corwin;A. Hammond;Milind Hegde;K. Matetski
中科院分区：
文献类型：
--
作者：
Ivan Corwin;A. Hammond;Milind Hegde;K. Matetski
在2002年，Johansson证明了Airy 2过程减去抛物线x的最大值几乎肯定在一个唯一的位置达到[Joh 03，猜想1.5]。10年后，Corwin和哈蒙德[CH 14，定理4.3]; Moreno弗洛雷斯，Quastel和Remenik [R 13];以及Pimentel [Pim 14]证明了这个结果。按比例缩放，Airy 2过程减去抛物线x 2，当从窄楔形初始数据初始化时，作为KPZ固定点的固定时间空间边缘出现。我们扩展这个最大化的唯一性结果的固定时间空间边缘的KPZ不动点时，从任何元素的一个非常广泛的初始数据初始化。这些结果都不排除在随机时间，KPZ不动点空间边际违反最大化唯一性的可能性。我们证明了，对于一个非常广泛的初始数据类，它是正概率，这样的时间的集合是非空的，并且，有条件地在这个事件，这一组几乎肯定有Hausdorff维数三分之二。在定向聚合物方面，这些时间的最大化非唯一性是在零温度聚合物测量不稳定的时刻，在该时刻，端点从一个位置跳到另一个。我们的分析依赖于Matetski，Quastel和Remenik在[MQR 21 b]中得到的KPZ不动点分布函数的精确公式，Dauvergne，Ortmann和Virág在[DOV 18]中构造的涉及Airy单的KPZ不动点的变分公式，以及Corwin和哈蒙德在[CH 14]中证明的Airy 2过程减去抛物线x 2的Brown Gibbs性质。
In 2002, Johansson conjectured that the maximum of the Airy2 process minus the parabola x is almost surely achieved at a unique location [Joh03, Conjecture 1.5]. This result was proved a decade later by Corwin and Hammond [CH14, Theorem 4.3]; Moreno Flores, Quastel and Remenik [FQR13]; and Pimentel [Pim14]. Up to scaling, the Airy2 process minus the parabola x 2 arises as the fixed time spatial marginal of the KPZ fixed point when initialized from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when initialized from any element of a very broad class of initial data. None of these results rules out the possibility that at random times, the KPZ fixed point spatial marginal violates maximizer uniqueness. We prove that, for a very broad class of initial data, it is with positive probability that the set of such times is non-empty, and that, conditionally on this event, this set almost surely has Hausdorff dimension two-thirds. In terms of directed polymers, these times of maximizer non-uniqueness are instants of instability in the zero temperature polymer measure—moments at which the endpoint jumps from one location to another. Our analysis relies on the exact formula for the distribution function of the KPZ fixed point obtained by Matetski, Quastel and Remenik in [MQR21b], the variational formula for the KPZ fixed point involving the Airy sheet constructed by Dauvergne, Ortmann and Virág in [DOV18], and the Brownian Gibbs property for the Airy2 process minus the parabola x 2 demonstrated by Corwin and Hammond in [CH14].