On the Exponent Governing the Correlation Decay of the Airy1 Process

We study the decay of the covariance of the Airy(1) process, A(1), a stationary stochastic process on 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 Cov(A(1)(0), A(1)(u)) = e(-(4/3+o(1))u3) as u -> infinity. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of 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.

Automated summary

This article studies the decay of covariance in the Airy(1) process, A(1), and determines its leading order term in the exponent. The proof combines probabilistic techniques and integrable probability estimates, with the upper bound utilizing the connection to planar exponential last passage percolation and the lower bound primarily using analytic methods and Fredholm determinant expressions.