  We present a proof of the power law decay of magnetic moment for the
$q$-state antiferromagnetic Potts model on the regular tree at the critical
temperature, and also justify that the exact exponent is $\frac{1}{2}$. Our
proof relies on the assumption of the uniqueness at the critical temperature,
which has been established for $q=3,4$, and for $q \ge 5$ with large degree. An
iterative contraction inequality is developed for independent interests.
