Law of iterated logarithms and fractal properties of the KPZ equation. (English) Zbl 1516.35573
Summary: We consider the Cole-Hopf solution of the \((1+1)\)-dimensional KPZ equation started from the narrow wedge initial condition. In this article we ask how the peaks and valleys of the KPZ height function (centered by time/24) at any spatial point grow as time increases. Our first main result is about the law of iterated logarithms for the KPZ equation. As time variable \(t\) goes to \(\infty\), we show that the limsup of the KPZ height function with the scaling by \(t^{1/3}(\log \log t)^{2/3}\) is almost surely equal to \((3/4\sqrt{2})^{2/3}\), whereas the liminf of the height function with the 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 fractal properties of the KPZ equation. Under exponential transformation of the time variable, we show that the peaks of KPZ height function mutate from being monofractal to multifractal, a property reminiscent of a similar phenomenon in Brownian motion (Theorem 1.4 in [D. Khoshnevisan et al., Ann. Probab. 45, No. 6A, 3697–3751 (2017; Zbl 1418.60081)]).
The proofs of our main results hinge on the following three key tools: (1) a multipoint composition law of the KPZ equation which can be regarded as a generalization of the two point composition law from (Proposition 2.9 in [I. Corwin et al., Ann. Probab. 49, No. 2, 832–876 (2021; Zbl 1467.60045)]), (2) the Gibbsian line ensemble techniques from [I. Corwin and A. Hammond, Invent. Math. 195, No. 2, 441–508 (2014; Zbl 1459.82117); Probab. Theory Relat. Fields 166, No. 1–2, 67–185 (2016; Zbl 1357.82040); Corwin et al., loc. cit.], and (3) the tail probabilities of the KPZ height function in short time and its spatiotemporal modulus of continuity. We advocate this last tool as one of our new and important contributions which might garner independent interest.
The proofs of our main results hinge on the following three key tools: (1) a multipoint composition law of the KPZ equation which can be regarded as a generalization of the two point composition law from (Proposition 2.9 in [I. Corwin et al., Ann. Probab. 49, No. 2, 832–876 (2021; Zbl 1467.60045)]), (2) the Gibbsian line ensemble techniques from [I. Corwin and A. Hammond, Invent. Math. 195, No. 2, 441–508 (2014; Zbl 1459.82117); Probab. Theory Relat. Fields 166, No. 1–2, 67–185 (2016; Zbl 1357.82040); Corwin et al., loc. cit.], and (3) the tail probabilities of the KPZ height function in short time and its spatiotemporal modulus of continuity. We advocate this last tool as one of our new and important contributions which might garner independent interest.
MSC:
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
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