Weak universality of global dynamics for the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model

We continue the study of weak universality for the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model initiated by Tzvetkov, Wang, and the author (2023). In this work, we establish weak universality in a regime where the associated Gibbs measure is mutually singular with respect to the underlying Gaussian measure. We first investigate the weak convergence of the Gibbs measures associated with macroscopic models to the fractional ${\mathrm{\Phi }}_3^4$-measure constructed by Tzvetkov, Wang, and the author (2023). We then obtain a structured representation for these Gibbs measures via the stochastic quantization method. Finally, we prove dynamical weak universality for the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model by using an extended second order expansion and exploiting the invariance of the Gibbs measures.