An upper-symbol approach to Φ4d limits of Bose gases
Abstract
We study the high-temperature derivation of Φ4d 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 ε=λη, 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 Φ42 derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local Φ42 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 use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert--Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.
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BibTeX
@misc{liang2026uppersymbol,
title = {{An upper-symbol approach to $Φ^4_d$ limits of Bose gases}},
author = {Hao Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26518},
eprint = {2609.26518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26518},
}
RIS
TY - UNPB
TI - An upper-symbol approach to Φ^4_d limits of Bose gases
AU - Hao Liang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.26518
UR - https://arxiv.org/abs/2609.26518
AB - We study the high-temperature derivation of Φ^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 ε=λ^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^4_2 derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local Φ^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 use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert--Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.
ER -