How flat is flat in random interface growth?

Abstract

Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on a large deviation rate function for the rescaled initial data, which arises naturally from the Hopf-Cole transformation. This allows us, in particular, to distinguish the domains of attraction of curved, flat, and Brownian initial data, and to identify the boundary between the curved and flat domains of attraction, which turns out to correspond to square root initial data. The distribution of the asymptotic one-point fluctuations is characterized by means of a variational formula written in terms of certain limiting processes (arising as subsequential limits of the spatial fluctuations of KPZ equation with narrow wedge initial data, as shown in [CH16]) which are widely believed to coincide with the Airy${}_2$ process. In order to identify these distributions for general initial data, we extend earlier results on continuum statistics of the Airy${}_2$ process to probabilities involving the process on the entire line. In particular, this allows us to write an explicit Fredholm determinant formula for the case of square root initial data.

Improved presentation and minor mistakes corrected. To appear in Trans. Am. Math. Soc

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Cited by in corpus (13)

• Stationary half-space last passage percolation
• KP governs random growth off a one dimensional substrate
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Stochastic growth in time dependent environments
• Upper tail decay of KPZ models with Brownian initial conditions
• KPZ fluctuations in finite volume
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• TASEP fluctuations with soft-shock initial data
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• TASEP with a general initial condition and a deterministically moving wall