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We prove that the log-Sobolev inequality holds for the two-dimensional $SU(N)$ Wilson lattice Yang--Mills measure with arbitrary fixed inverse coupling $β$. The log-Sobolev constant depends on $β$ and N, but is uniform in the volume of the underlying lattice. The proof relies on a two-scale disintegration argument.
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Piazza et al. (2026) studied this question.
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