Tails of the endpoint distribution of directed polymers

We prove that the random variable $\text{ct}={\mathrm{arg\hspace{0.17em}max}}_{t\in \text{rr}}\{\text{aip}(t)-t^2\}$ has tails which decay like $e^{-c{t}^3}$. The distribution of $\text{ct}$ is a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions for large time or temperature.