Fluctuations in the Potts spin glass model at high temperature

We study the fluctuations of the free energy of the Potts spin glass with $\kappa$ colors at high temperature. The disorder is symmetric with unit variance and finite fourth moment. For configurations with a balanced, or nearly balanced, color profile, the log-partition function satisfies a central limit theorem whenever ${\beta }^2<{\alpha }_c(\kappa )$. Here $\kappa {\alpha }_c(\kappa )$ is the critical inverse temperature of the Curie-Weiss-Potts model. A comparison of color profiles transfers this limit to the unconstrained model for $\beta <{\beta }_{\max }(\kappa )$, an explicit threshold. In the same range, the magnetizations of two replicas and their overlap matrix satisfy a joint quenched central limit theorem with independent Gaussian limits. The fourth cumulant of the disorder enters the limiting mean and variance explicitly.