On the exponent governing the correlation decay of the Airy$_1$ process

We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy$_1$ process together with the FKG inequality.