  We study the asymptotic behavior of quantized Ding functionals along Bergman
geodesic rays and prove that the slope at infinity can be expressed in terms of
Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we
introduce a new algebro-geometric stability on Fano manifolds and show that the
existence of anti-canonically balanced metrics implies our stability. The
relation between our stability and others is also discussed. As another
application of the slope formula, we get the lower bound estimate on the Calabi
like functionals on Fano manifolds.
