  We prove an optimal $\Omega(n^{-1})$ lower bound for modified log-Sobolev
(MLS) constant of the Glauber dynamics for anti-ferromagnetic two-spin systems
with $n$ vertices in the tree uniqueness regime. Specifically, this optimal MLS
bound holds for the following classes of two-spin systems in the tree
uniqueness regime:
  $\bullet$ all strictly anti-ferromagnetic two-spin systems (where both edge
parameters $\beta,\gamma<1$), which cover the hardcore models and the
anti-ferromagnetic Ising models;
  $\bullet$ general anti-ferromagnetic two-spin systems on regular graphs.
  Consequently, an optimal $O(n\log n)$ mixing time holds for these
anti-ferromagnetic two-spin systems when the uniqueness condition is satisfied.
These MLS and mixing time bounds hold for any bounded or unbounded maximum
degree, and the constant factors in the bounds depend only on the gap to the
uniqueness threshold. We prove this by showing a boosting theorem for MLS
constant for distributions satisfying certain spectral independence and
marginal stability properties.
