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: \(\begin{aligned} \liminf _{R\rightarrow \infty }\frac{ \max _{|x|\le R}\left( \log u(t\,,x) + \frac{x^2}{2t}\right) }{(\log R)^{2/3}} \ge \frac{1}{4}\left( \frac{t}{2}\right) ^{1/3}, \quad \text {a.s.} \end{aligned}\)
