论文标题
KPZ固定点的迭代对数的长时间和短期定律
Long and short time laws of iterated logarithms for the KPZ fixed point
论文作者
论文摘要
我们考虑从一般初始数据开始的KPZ固定点。在本文中，我们研究了KPZ固定点的大峰值的增长，即时间$ t $变为$ \ infty $，而当$ t $接近$ 1 $时，我们就以空间点的$ 0 $ 0 $ 0 $ 0。我们证明，对于非常广泛的初始数据类别，作为$ t \ to \ infty $，当用$ t^{1/3}（\ log \ log \ log \ log t）^{2/3}^缩放时，kpz固定点高度功能的limsup几乎肯定平等一个常数。常数的值为$（3/4）^{2/3} $或$（3/2）^{2/3} $，具体取决于初始数据分别为非兰德或布朗尼人。此外，我们表明KPZ固定点接近$ t = 1 $的增量允许迭代对数的短期定律。更准确地说，随着时间的增加$ΔT：= t-1 $降至$ 0 $，对于包括Brownian数据初始数据在内的大量初始数据，我们表明，高度的Limsup会增加KPZ固定点$ 1 $ $ 1 $，当时$（ΔT）^{1/3}^{1/3}（1/3}（1/3}） $（3/2）^{2/3} $。
We consider the KPZ fixed point starting from a general class of initial data. In this article, we study the growth of the large peaks of the KPZ fixed point at a spatial point $0$ when time $t$ goes to $\infty$ and when $t$ approaches $1$. We prove that for a very broad class of initial data, as $t\to \infty$, the limsup of the KPZ fixed point height function when scaled by $t^{1/3}(\log\log t)^{2/3}$ almost surely equals a constant. The value of the constant is $(3/4)^{2/3}$ or $(3/2)^{2/3}$ depending on the initial data being non-random or Brownian respectively. Furthermore, we show that the increments of the KPZ fixed point near $t=1$ admits a short time law of iterated logarithm. More precisely, as the time increments $Δt :=t-1$ goes down to $0$, for a large class of initial data including the Brownian data initial data, we show that limsup of the height increments the KPZ fixed point near time $1$ when scaled by $(Δt)^{1/3}(\log\log (Δt)^{-1})^{2/3}$ almost surely equals $(3/2)^{2/3}$.
