Lower bound on spatial asymptotic of parabolic Anderson model with narrow wedge initial condition

Let $\{u(t\,,x): (t,x)\in (0, \infty)\times \mathbb{R}\}$ be the solution to parabolic Anderson model with narrow wedge initial condition. Using the association property of parabolic Anderson model, we establish a lower bound on spatial asymptotic of the solution: \begin{align*} \liminf_{R\to\infty}\frac{ \max_{|x|\leq R}\left(\log u(t\,,x) + \frac{x^2}{2t}\right)}{(\log R)^{2/3}} \geq \frac14\left(\frac{t}2\right)^{1/3}, \quad \text{a.s.} \end{align*}