Nonuniversal Exponents in Interface Growth

Abstract

We report on an extensive numerical investigation of the Kardar-Parisi-Zhang equation describing non-equilibrium interfaces. Attention is paid to the dependence of the growth exponent β on the details of the distribution of noise p(ξ). All distributions considered are delta-correlated in space and time, and have finite cumulants. We find that β becomes progressively more sensitive to details of the distribution with increasing dimensionality. We discuss the implications of these results for the universality hypothesis.PACS numbers: 05.40.+j, 68.35.R Typeset using REVT E X 1

Universality of the directed polymer model

“…In dimensions higher then 2, there are only numerical estimates for the values of the exponents, and in the three and four dimensional cases, the accepted values for the exponents are ν ≃ 0.62, ω ≃ 0.24, and ν ≃ 0.59, ω ≃ 0.18 respectively [10,11]. However, these numerical estimates were obtained for random values taken from Gaussian distribution, while non-Gaussian distributions, though preserving the space exponents [11], yield lower estimates for the energy exponents [5,11]. The dependence of the estimated energy exponents on the form of the probability distribution suggests non-uniersality of the model [5], and a break of the scaling relation.…”

“…This article presents results of numerical simulations for the three and four dimensional cases studied in [5], but for larger lattices. The results show that the estimates presented in [5] are indeed below the asymptotic values of the exponents.…”

“…The random environment is usually simulated by a lattice whose bonds (or sites) are assigned with random values, and it is generally believed that the characteristics of the optimal path are universal, i.e. apart from very special cases, the exponents depend only on the dimension of the lattice and not on the details of the lattice structure, or the type of randomness associated with it.The universality hypothesis was recently challenged by numerical results [5] which showed that for dimensions higher than 2, the values of the estimated exponents do depend on the type of the probability distribution which is associated with the random lattice. In particular, nonGaussian distributions yield lower values for the exponents compared to Gaussian distribution values, which are the persumed universal values.…”

“…Two of these articles [6,7] independently suggested that the estimates presented in [5] are disturbed by directed percolation effects. However, none of these opposing articles presented any alternative numerical estimates to those presented in [5].This article presents results of numerical simulations for the three and four dimensional cases studied in [5], but for larger lattices. The results show that the estimates presented in [5] are indeed below the asymptotic values of the exponents.…”

“…However, none of these opposing articles presented any alternative numerical estimates to those presented in [5].…”