On the Exponent Governing the Correlation Decay of the hbox {Airy}_1 Process

Abstract

We study the decay of the covariance of the hbox {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 urightarrow 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 hbox {Airy}_1 process together with the FKG inequality.