Phase transitions for fractional $\Phi^3_d$ on the torus

Abstract

We consider the fractional $\Phi^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-\Delta)^\alpha$, and show a phase transition at $d=3\alpha$. More precisely, in a regular regime $d<3\alpha$, one can construct and normalise this measure, and obtain a measure which is absolutely continuous with respect to the Gaussian free field $\mu$. At $d=3\alpha$, the behaviour depends on the size $|\sigma|$ of the nonlinearity: for $|\sigma|\ll1$, the measure exists, but is singular with respect to $\mu$, whereas for $|\sigma|\gg1$, the measure is not normalisable. This generalises a result of Oh, Okamoto, and Tolomeo (2025) on the $\Phi^3_3$-measure.