Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and ${\mathrm{Airy}}_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, 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 $\langle L\left(t\right)\rangle =L_0+\omega t^{\gamma }$, 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 ${\mathrm{Airy}}_2$ only for $\omega \gg 1$, for a given $\gamma$, or when $\gamma =1$, for a fixed $\omega$. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

• Received 19 October 2018

DOI:https://doi.org/10.1103/PhysRevE.99.032140