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 all, including unbounded, maximum degree $Delta$. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack $deltain(0,1)$: $bullet$ $C(delta) n^2log n$ mixing time for the hardcore model with fugacity $lambdale (1-delta)lambda_c(Delta)= (1-delta)frac{(Delta - 1)^{Delta - 1}}{(Delta - 2)^Delta}$; $bullet$ $C(delta) n^2$ mixing time for the Ising model with edge activity $betainleft[frac{Delta-2+delta}{Delta-delta},frac{Delta-delta}{Delta-2+delta}right]$; where the maximum degree $Delta$ may depend on the number of vertices $n$, and $C(delta)$ depends only on $delta$. 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.