Pitfalls on the determination of the universality class of radial clusters

TL;DR

This study reveals that the measured growth exponent of radial clusters depends on the choice of origin for interface width calculation, highlighting a potential pitfall in classifying universality classes of such systems.

Contribution

The paper demonstrates how the evaluation of the growth exponent varies with the origin choice, clarifying discrepancies in previous measurements and emphasizing the importance of fluctuation scales.

Findings

The growth exponent depends on the origin used for interface width measurement.

Using the initial seed yields the KPZ exponent of 1/3.

Border CM fluctuations grow faster than interface fluctuations.

Abstract

The self-affinity of growing systems with radial symmetry, from tumors to grain-grain displacement, has devoted increasing interest in the last decade. In this work, we analyzed features about the interface scaling of these clusters through large scale simulations (up to $3\times 10^7$ particles) of two-dimensional growth processes with special emphasis on the off-lattice Eden model. The central objective is to discuss an important pitfall associated to the evaluation of the growth exponent $\beta$ of these systems. We show that the $\beta$ value depends on the choice of the origin used to determine the interface width. We considered two strategies frequently used. When the width is evaluated in relation to the center of mass (CM) of the border, the exponent obtained for the Eden model was $\beta_{CM}=0.404\pm0.013$, in very good agreement with previous reported values. However, if the border CM is replaced by the initial seed position (a static origin), the exponent $\beta_0=0.333\pm 0.010$, in complete agreement with the KPZ value $\beta_{KPZ}=1/3$, was found. The difference between $\beta_{CM}$ and $\beta_{0}$ was explained through the border CM fluctuations that grow faster than the overall interface fluctuations. Indeed, we show that the exponents $\beta_0$ and $\beta_{CM}$ characterize large and small wavelength fluctuations of the interface, respectively. These finds were also observed in three distinct lattice models, in which the lattice-imposed anisotropy is absent.