Before revision:  Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs  .
After revision:  Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs . In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting .