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

Abstract

AbstractWe study the decay of the covariance of the $$\hbox {Airy}_1$$ Airy 1 process, $${{\mathcal {A}}}_1$$ A 1 , a stationary stochastic process on $$\mathbb {R}$$ 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 $${{\,\textrm{Cov}\,}}({{\mathcal {A}}}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$$ Cov ( A 1 ( 0 ) , A 1 ( u ) ) = e - ( 4 3 + o ( 1 ) ) u 3 as $$u\rightarrow \infty $$ u → ∞ . The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $$\mathcal{A}_1$$ 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 $$\hbox {Airy}_1$$ Airy 1 process together with the FKG inequality.