Universal fluctuations in radial growth models belonging to the KPZ universality class

Abstract

PACS 81.15.Aa -Theory and models of film growth PACS 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion PACS 89.75.Da -Systems obeying scaling laws Abstract. -We investigate the radius distributions (RD) of surfaces obtained with large-scale simulations of radial clusters that belong to the KPZ universality class. For all investigated models, the RDs are given by the Tracy-Widom distribution of the Gaussian unitary ensemble, in agreement with the conjecture of the KPZ universality class for curved surfaces. The quantitative agreement was also confirmed by two-point correlation functions asymptotically given by the covariance of the Airy2 process. Our simulation results fill the last lacking gap of the conjecture that had been recently verified analytically and experimentally.

Origins of scaling corrections in ballistic growth models

“…(1)], Γ is a nonuniversal constant associated to the amplitude of the interface fluctuations, β is the growth exponent, and χ is a stochastic quantity given by Tracy-Widom [13] distributions. This conjecture was confirmed in distinct KPZ systems [14][15][16][17][18][19] besides exact solutions of KPZ equation [20][21][22][23]. Recent numerical simulations have shown that the KPZ ansatz can be generalized to 2+1 [24][25][26] and higher [27] dimensions, but the exact forms of the asymptotic distributions of χ are yet not known.…”

A KPZ Cocktail-Shaken, not Stirred...

“…We mention that, aside from some early works [101,102,103], and more recently [104], there has been little direct effort on numerical integration of 2d KPZ equation in polar coordinates; see, too [105,106,107]. Indeed, all work on this subclass, aside from radial Eden model simulations [108,109], have resorted to difficult, somewhat frustrating pt-pt simulations of various KPZ/DPRM models [110,65] in what is, effectively, constrained wedge geometries. The frustration arises because, in contrast to simulations for the flat KPZ subclass where all substrate points contribute to the ensemble average, the pt-pt Monte Carlo yields a few datum only per realization.…”

Dimensional fragility of the Kardar–Parisi–Zhang universality class

“…Thus, the exact asymptotic height distribution function has been very recently obtained for d = 1 [13][14][15]: it is given by the largest-eigenvalue distribution of large random matrices in the Gaussian unitary (GUE) (orthogonal, GOE) ensemble, the Tracy-Widom (TW) distribution, for globally curved (flat) interfaces, as proposed in [16], see reviews in [17,18]. Besides elucidating fascinating connections with probabilistic and exactly solvable systems, these results are showing that, not only are the critical exponent values common to members of this universality class, but also the distribution functions and limiting processes are shared by discrete models and continuum equations [19], and by experimental systems, from turbulent liquid crystals [20] to drying colloidal suspensions [21].In view of the success for d = 1 (1D) substrates, a natural important step is to assess the behavior of the KPZ universality class when changing space dimension, analogous to e.g. the experimental change from 2D to 1D behavior for ferromagnetic nanowires, that nonetheless occurs within the creeping-domain-wall class [22].…”