Anticanonically balanced metrics on Fano manifolds

We show that if a Fano manifold has discrete automorphism group and admits a polarized K\"ahler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the K\"ahler-Einstein metric. Our proof is based on a simplification of Donaldson's proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin-Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson's iterations in the anticanonical setting.