Sharp phase transition in the grand canonical ${\mathrm{\Phi }}^3$ measure at critical chemical potential

We study the phase transition and critical phenomenon for the grand canonical ${\mathrm{\Phi }}^3$ measure in two-dimensional Euclidean quantum field theory. The study of this measure was initiated by Jaffe, Bourgain, and Carlen–Fröhlich–Lebowitz, primarily in regimes far from criticality. We identify a critical chemical potential and show that the measure exhibits a sharp phase transition at this critical threshold. At the critical threshold, the analysis is based on establishing the correlation decay of the Gaussian fluctuations in the partition function, combined with a coarse-graining argument to show divergence of the maximum of an approximating Gaussian process.