Circular Kardar-Parisi-Zhang interfaces evolving out of the plane

Area of Science:

• Statistical Physics
• Complex Systems
• Surface Growth Models

Background:

• Standard 1D Kardar-Parisi-Zhang (KPZ) interfaces on flat surfaces exhibit specific statistical properties.
• Circular interfaces on a plane show Gaussian Unitary Ensemble (GUE) Tracy-Widom (TW) height distributions and Airy_{2} spatial covariance.
• The impact of evolving background spaces on these interface statistics remains largely unexplored.

Purpose of the Study:

• To investigate the statistical properties of 1D KPZ interfaces evolving on curved substrates with expanding sizes.
• To determine how substrate expansion rate and curvature influence interface height distributions and spatial covariances.
• To generalize the understanding of KPZ universality classes beyond flat surfaces.

Main Methods:

• Extensive numerical simulations of 1D KPZ models on various surfaces of revolution.
• Analysis of interface height distributions (HD) and spatial covariances.
• Mathematical modeling of substrate size enlargement (〈L(t)〉=L_{0}+ωt^{γ}) and its competition with correlation length (ξ≃ct^{1/z}).

Main Results:

• Interface statistics depend critically on the interplay between substrate expansion rate (γ) and correlation length exponent (1/z).
• For γ > 1/z, GUE height distributions are observed, but spatial covariances deviate from Airy_{2} unless ω is large or γ=1.
• For γ < 1/z, Gaussian height distributions emerge in a correlated regime.
• A continuous transition between Gaussian and GUE distributions occurs at γ = 1/z, with Gaussian Symplectic Ensemble (GSE) TW distribution observed near ω/c ≈ 10.

Conclusions:

• The background space significantly impacts the asymptotic statistics of 1D KPZ interfaces.
• The competition between substrate enlargement and intrinsic interface dynamics dictates the universality class.
• These findings extend the known universality of KPZ models and highlight the role of geometric constraints.