Determine the maximal shrinking rate of the interaction range

Determine how rapidly the interaction range \(\varepsilon\) can shrink as \(\lambda\to0\) while preserving the relative free-energy and reduced-density-matrix convergence limits for the high-temperature Bose-gas-to-Hartree and \(\Phi^4_d\) derivations.

Background

The paper studies grand-canonical Bose gases on two- and three-dimensional tori in the high-temperature limit $\lambda\to0$, with interaction range $\varepsilon=\lambda^\eta$. It proves convergence to renormalized Hartree Gibbs measures for every fixed $0<\eta<1/2$ in two dimensions and $0<\eta<1/26$ in three dimensions, including relative free-energy convergence and Hilbert–Schmidt convergence of every fixed-order reduced density matrix.

The authors motivate a sharper range by comparing $\varepsilon$ with the typical interparticle distance. In the stated scaling, the dilute regime corresponds essentially to $\eta>1/2$, whereas the proven two-dimensional range approaches $\eta=1/2$ from below. The precise maximal rate at which $\varepsilon$ may shrink while retaining the convergence results is not determined by the paper.