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We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equationstarted from the narrow wedge initial condition. In this article, we ask howthe peaks and valleys of the KPZ height function (centered by time/24) at anyspatial point grow as time increases. Our first main result is about the law ofiterated logarithms for the KPZ equation. As time variable $t$ goes to$\infty$, we show that the limsup of the KPZ height function with the scalingby $t^{1/3}(\log\log t)^{2/3}$ is almost surely equal to$(\frac{3}{4\sqrt{2}})^{2/3}$ whereas the liminf of the height function withthe scaling by $t^{1/3}(\log\log t)^{1/3}$ is almost surely equal to$-6^{1/3}$. Our second main result concerns with the macroscopic fractalproperties of the KPZ equation. Under exponential transformation of the timevariable, we show that the peaks of KPZ height function mutate from beingmonofractal to multifractal, a property reminiscent of a similar phenomenon inBrownian motion [Khoshnevisan-Kim-Xiao 17, Theorem 1.4].  The proofs of our main results hinge on the following three key tools: (1) amulti-point composition law of the KPZ equation which can be regarded as ageneralization of the two point composition law from [Corwin-Ghosal-Hammond 19,Proposition 2.9], (2) the Gibbsian line ensemble techniques from[Corwin-Hammond 14, Corwin-Hammond 16, Corwin-Ghosal-Hammond 19] and, (3) thetail probabilities of the KPZ height function in short time and itsspatio-temporal modulus of continuity. We advocate this last tool as one of ournew and important contributions which might garner independent interest.

Law of Iterated Logarithms and Fractal Properties of the KPZ Equation