We study the competition interface between two clusters growing over a random vacant sector of the plane in a simple setup which allows us to perform formal computations and obtain analytical solutions. We demonstrate that a phase transition occurs for the asymptotic inclination of this interface when the final macroscopic shape goes from curved to noncurved. In the first case it is random while in the second one it is deterministic. We also show that the flat case (stationary growth) is a critical point for the fluctuations: for curved and flat final profiles the fluctuations are in the Kardar-Parisi-Zhang (KPZ) scale (2/3); for noncurve final profile the fluctuations are in the same scale of the fluctuations of the initial conditions, which in our model are Gaussian (1/2).
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