On the exponent governing the correlation decay of the Airy₁ process

Abstract

We study the decay of the covariance of the Airy₁ process, A₁, 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 (A₁(0), A₁(u)) = e^{-(4/3+o(1))u3} as u→∞. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of A₁ 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 Airy₁ process together with the FKG inequality.