Long and short time laws of iterated logarithms for the KPZ fixed point

Abstract

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 $\hat{I}”t:=t-1$ 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 $(\hat{I}”t{)}^{1/3}(\log \log (\hat{I}”t{)}^{-1}{)}^{2/3}$ almost surely equals $(3/2{)}^{2/3}$.

35 pages, no figures