Motivated by weak universality problems in singular SPDEs, we consider a class of approximations to 2D defocusing (deterministic) wave equation with singular random initial data, as well as their corresponding Gibbs measures. We give a sufficient and almost necessary condition for the convergence of Gibbs measures within this class of approximation. Interestingly, this condition turns out to be strictly stronger than that for the global convergence of the wave dynamics. The main ingredient in treating these measures is a variational method recently developed by Barashkov and Gubinelli.

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