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Changeover phenomenon in randomly colored Potts models

A hybrid Potts model where a random concentration $p$ of the spins assume $q_0$ states and a random concentration $1-p$ of the spins assume $q>q_0$ states is introduced. It is known that when the system is homogeneous, with an integer spin number $q_0$ or $q$, it undergoes a second or a first order transition, respectively. It is argued that there is a concentration $p^\ast$ such that the transition nature of the model is changed at $p^\ast$. This idea is demonstrated analytically and by simulations for two different types of interaction: the usual square lattice nearest neighboring and mean field all-to-all. Exact expressions for the second order critical line in concentration-temperature parameter space of the mean field model together with some other related critical properties, are derived.