  In the Potts spin glass model, inspired by the symmetry argument in
[arXiv:2310.06745] for the constrained free energy, we study the free energy
with self-overlap correction. Similarly, we simplify the Parisi-type formula,
originally an infimum over matrix-valued paths, into an optimization over
real-valued paths, which can be interpreted as quantile functions of
probability measures on the unit interval. Minimizers are usually called Parisi
measures. Using results in [arXiv:2311.10446] generalizing the Auffinger-Chen
convexity argument, we deduce the uniqueness of the Parisi measure.
  The approach in [arXiv:2310.06745] is to perturb the covariance function of
the Hamiltonian into one associated with the generic model, prove the results
there, and then reduce the perturbation. Here, we choose to perturb the model
by adding an external field parametrized by Ruelle probability cascades. This
is used in the Hamilton-Jacobi equation approach and we directly apply results
in [arXiv:2311.08980].
