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Pitfalls on the determination of the universality class of radial clusters

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 $β$ of these systems. We show that the $β$ 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 $β_{CM}=0.404\pm0.013$, 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 $β_0=0.333\pm 0.010$, in complete agreement with the KPZ value $β_{KPZ}=1/3$, was found. The difference between $β_{CM}$ and $β_{0}$ was explained through the border CM fluctuations that grow faster than the overall interface fluctuations. Indeed, we show that the exponents $β_0$ and $β_{CM}$ 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.