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

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.