2026/07/25 by Phan Thành Nam, Zhilin Yang, Xiangchan Zhu
#math-ph #math.AP #math.MP #math.PR
We derive the P(Φ)2 measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the Φ2p2 measure is formulated in the grand-canonical ensemble with a general symmetric p-body interaction potential which is not required to have a factorized form. Unlike in the Φ42 case investigated by Fröhlich--Knowles--Schlein--Sohinger in \citeFKSS25, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading p-body interaction to control all lower-order terms generated by the Wick counterterms.