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Using the Polyakov-loop effective action, we investigate the structure ofspontaneously broken $\mathbb{Z}_N$ symmetry in the deconfinement vacuum in theSU($N$) Yang-Mills theory with finite $N$. First, we examine the Polyakov-loopfluctuation around a $\mathbb{Z}_N$-broken vacuum and calculate the spatialcorrelation of its phase variable. We show that the phase variable of thePolyakov loop becomes a Nambu-Goldstone mode in the large-$N$ limit. Second, weestimate the global vacuum-to-vacuum transition rate in a finite-volume domainof the quark-gluon plasma. Based on our estimation, we state that somethreshold volume exists, a domain larger than which is stable, and vice versa.Identifying the threshold as the lower bound of a stable center domain volume,we find the typical volume scale of center domains.

$\mathbb{Z}_N$ structure of deconfinement vacuum in SU($N$) Yang-Mills theory: emergence of Nambu-Goldstone mode in large-$N$ limit