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

Abstract

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 $\text{Cov}({\mathcal{A}}_1(0),{\mathcal{A}}_1(u))=e^{-(\frac{4}{3}+o(1))u^3}$ 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.

51 pages, 5 figures, LaTeX

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Cited by in corpus (3)

• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Optimal tail estimates in $\beta$-ensembles and applications to last passage percolation
• Uniqueness, mixing, and optimal tails for Brownian line ensembles with geometric area tilt