An upper-symbol approach to $Φ^4_d$ limits of Bose gases

Abstract: We study the high-temperature derivation of $Φ<sup>4_d$ measures from grand-canonical Bose gases on two- and three-dimensional tori. Building on the upper-symbol variational approach of the author's earlier work, we use relative entropy to give a simple, unified proof of relative free-energy convergence and Hilbert--Schmidt convergence of all fixed-order rescaled reduced density matrices to their Hartree counterparts. For interaction ranges $\varepsilon=λ<sup>η$, where $λ$ is the inverse temperature, these results hold for every fixed $0<η<1/2$ in two dimensions and $0<η<1/26$ in three. In two dimensions, we improve the result of Jougla and Rougerie (2026) and the local $Φ<sup>4_2$ derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local $Φ<sup>4_2$ derivation valid for every fixed exponent $0<η<1/2$, allowing polynomial exponents arbitrarily close to the diluteness threshold from below. In three dimensions, we strengthen the weak convergence at finitely many orders established by Nam, R.~Zhu and X.~Zhu Theorem~2.8 to Hilbert--Schmidt convergence at every fixed order.