论文标题
稳定和混乱的最后一次通道渗透
Stability and chaos in dynamical last passage percolation
论文作者
论文摘要
许多复杂的统计机械模型具有复杂的能量景观。基态或最低的能量状态位于最深山谷的底部。在自旋眼镜和高斯聚合物等示例中，有很多山谷。近地状态（位于山谷基地）的丰度表明了混乱的现象，当模型的疾病略微干扰时，基态在这种现象下会深刻地改变。在本文中，我们计算了在Kardar-Parisi-Zhang [kpz]普遍性中，在典型模型的动态表现中控制混乱开始的关键指数，Brownian Last Passage Percolation [LPP]。在这种静态形式的模型中，半分化聚合物通过布朗尼噪声前进，它们在旅途中遇到的白噪声的积分所提供的能量。鉴于其终点，基态是一种极值的大地测量。我们通过在Ornstein-Uhlenbeck流下进化该疾病来扰动Brownian LPP。我们证明，对于长度$ n $的聚合物，在关键时间$ n^{ - 1/3} $的关键时间见证了一个尖锐的阶段过渡。实际上，在$ n $的给定距离$ n $的$ n $时，Geodesics在零和$ t> 0 $之间的重叠将是$ n $当$ t \ ll n^{ - 1/3} $时；当$ t \ gg n^{ - 1/3} $时，订单较小。我们希望该指数在许多接口模型中共享。因此，目前的工作阐明了KPZ类的动态方面。它基于最近的几个进步。其中包括Chatterjee的谐波分析理论[CHA14]高斯空间中超浓度和混乱的等效性；对伴侣论文中布朗尼LPP的静态景观几何形状的精致理解[GH20]；在后者的基础上，在[CHH19]中，测量能曲线与布朗运动的强烈比较估计值。
Many complex statistical mechanical models have intricate energy landscapes. The ground state, or lowest energy state, lies at the base of the deepest valley. In examples such as spin glasses and Gaussian polymers, there are many valleys; the abundance of near-ground states (at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the model's disorder is slightly perturbed. In this article, we compute the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semi-discrete polymers advance through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is a geodesic, of extremal energy given its endpoints. We perturb Brownian LPP by evolving the disorder under an Ornstein-Uhlenbeck flow. We prove that, for polymers of length $n$, a sharp phase transition marking the onset of chaos is witnessed at the critical time $n^{-1/3}$. Indeed, the overlap between the geodesics at times zero and $t > 0$ that travel a given distance of order $n$ will be shown to be of order $n$ when $t\ll n^{-1/3}$; and to be of smaller order when $t\gg n^{-1/3}$. We expect this exponent to be shared among many interface models. The present work thus sheds light on the dynamical aspect of the KPZ class; it builds on several recent advances. These include Chatterjee's harmonic analytic theory [Cha14] of equivalence of superconcentration and chaos in Gaussian spaces; a refined understanding of the static landscape geometry of Brownian LPP developed in the companion paper [GH20]; and, underlying the latter, strong comparison estimates of the geodesic energy profile to Brownian motion in [CHH19].
