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 $⟨L(t)⟩=L_0+\omega t^{\gamma }$, while their mean height $⟨h⟩$ increases as usual [$⟨h⟩\sim t$]. 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.

arXiv comment8 pages and 8 figures, including a Suppl. Material

References in corpus (29)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• The one-dimensional KPZ equation: an exact solution and its universality
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Growing interfaces uncover universal fluctuations behind scale invariance
• An exact solution for the KPZ equation with flat initial conditions
• Spatial correlations of the 1D KPZ surface on a flat substrate
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Directed polymer near a hard wall and KPZ equation in the half-space
• Kardar-Parisi-Zhang universality class in 2+1 dimensions: Universal geometry-dependent distributions and finite-time corrections
• Universal aspects of curved, flat & stationary-state Kardar-Parisi-Zhang statistics
• Universal fluctuations in radial growth models belonging to the KPZ universality class
• Universal fluctuations in KPZ growth on one-dimensional flat substrates
• Directed random polymers via nested contour integrals
• Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation
• Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth
• Interface fluctuations for deposition on enlarging flat substrates
• Kardar-Parisi-Zhang Interfaces with Inward Growth
• Topology and the Kardar-Parisi-Zhang universality class
• Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Stochastic growth equations on growing domains
• The circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer
• Probability distributions for directed polymers in random media with correlated noise
• Radial Restricted Solid-on-Solid and Etching Interface Growth Models

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• KPZ fluctuations in finite volume
• Surface growth on treelike lattices and the upper critical dimension of the KPZ class
• Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time
• One-point height fluctuations and two-point correlators of $(2+1)$ cylindrical KPZ systems
• Height distributions in interface growth: The role of the averaging process
• Numerical Simulation and the Universality Class of the KPZ Equation for Curved Substrates
• Dimensional crossover in Kardar-Parisi-Zhang growth