Rapid mixing of Glauber dynamics via spectral independence for all degrees

We prove an optimal
$\Omega(n^{-1})$
lower bound on the spectral gap of Glauber dynamics for anti-ferromagnetic two-spin systems with
$n$
vertices in the tree uniqueness regime. This spectral gap holds for any, including unbounded, maximum degree Δ. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack
$\delta\in(0,1)$
: •
$C(\delta)n^{2}\log n$
mixing time for the hardcore model with fugacity
$\leq(1-\delta)\lambda_{c}(\Delta)=(1-\delta)\frac{(\Delta-1)^{\Delta-1}}{(\Delta-2)^{\Delta}}$
; •
$C(\delta)n^{2}$
mixing time for the Ising model with edge activity
$\beta\in[\frac{\Delta-2+\delta}{\Delta-\delta},\frac{\Delta-\delta}{\Delta-2+\delta}]$
; where the maximum degree Δ may depend on the number of vertices n, and C(δ) depends only on δ. Our proof is built upon the recently developed connections between the Glauber dynamics for spin systems and the high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called the complete spectral independence, and use a novel Markov chain called the field dynamics to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.