Tail decay for the distribution of the endpoint of a directed polymer

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects the density function of $\tcal$ to the Hastings-McLeod solution of the second Painlev\'e equation, we prove that as $t\rightarrow\infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.