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

TL;DR

This paper investigates how the background surface shape affects the statistical properties of circular KPZ interfaces, revealing a rich variety of height distributions and covariances depending on substrate growth and geometry.

Contribution

It introduces a comprehensive numerical analysis of 1D KPZ models on evolving curved surfaces, showing how background geometry influences height distribution types and spatial covariances.

Findings

GUE Tracy-Widom distribution occurs for fast-enlarging substrates with gamma > 1/z

Gaussian distributions dominate for slow-enlarging substrates with gamma < 1/z

Interpolating distributions appear at the critical case gamma = 1/z

Abstract

Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a cup, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as $\langle L(t) \rangle = L_0+\omega t^{\gamma}$, while their mean height $\langle h \rangle$ increases as usual [$\langle h \rangle\sim t$]. We show that the competition between the $L$ enlargement and the correlation length ($\xi \simeq c t^{1/z}$) plays a key role in the asymptotic statistics of the interfaces. While systems with $\gamma>1/z$ have HDs given by GUE and the interface width increasing as $w \sim t^{\beta}$, for $\gamma<1/z$ the HDs are Gaussian, in a correlated regime where $w \sim t^{\alpha \gamma}$. For the special case $\gamma=1/z$, a continuous class of distributions exists, which interpolate between Gaussian (for small $\omega/c$) and GUE (for $\omega/c \gg 1$). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for $\omega/c \approx 10$. Despite the GUE HDs for $\gamma>1/z$, the spatial covariances present a strong dependence on the parameters $\omega$ and $\gamma$, agreeing with Airy$_2$ only for $\omega \gg 1$, for a given $\gamma$, or when $\gamma=1$, for a fixed $\omega$. These results considerably generalize our knowledge on the 1D KPZ systems, unveiling the importance of the background space in their statistics.