Phase transitions for fractional $Φ^3_d$ on the torus
Niko Nikov
TL;DR
This work establishes a phase transition for the fractional $\Phi^3_d$ measure on the $d$-torus with a Gaussian free field of covariance $(1-\Delta)^{-\alpha}$, occurring at the critical dimension $d=3\alpha$. By combining Wick renormalisation, a mass taming mechanism, and the Boué-Dupuis variational formula—including a second renormalisation in the critical regime—the authors construct and analyse truncated Gibbs measures, proving uniform exponential integrability, tightness, and convergence in subcritical and some critical regimes, as well as singularity or non-normalisability phenomena at criticality depending on the nonlinearity strength $|\sigma|$. In the subcritical and regular regimes ($d<3\alpha$ and $d=3\alpha$ with small $|\sigma|$), the limiting measure exists and is absolutely continuous with respect to the reference Gaussian $\mu$ (or singular at criticality), while in the strongly nonlinear critical regime the measures fail to be normalisable, illustrating a sharp phase transition. The results generalise the $\Phi^3_3$ case of Oh, Okamoto, and Tolomeo (2025) to fractional dimensions and establish a robust probabilistic framework for fractional Gibbs measures with implications for stochastic quantisation and long-time dynamics of related fractional SPDEs.
Abstract
We consider the fractional $Φ^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-Δ)^α$, and show a phase transition at $d=3α$. More precisely, in a regular regime $d<3α$, one can construct and normalise this measure, and obtain a measure which is absolutely continuous with respect to the Gaussian free field $μ$. At $d=3α$, the behaviour depends on the size $|σ|$ of the nonlinearity: for $|σ|\ll1$, the measure exists, but is singular with respect to $μ$, whereas for $|σ|\gg1$, the measure is not normalisable. This generalises a result of Oh, Okamoto, and Tolomeo (2025) on the $Φ^3_3$-measure.