Log-Sobolev inequality for the \(\varphi^4_2\) and \(\varphi^4_3\) measures. (English) Zbl 07818724
Summary: The continuum \(\varphi^4_2\) and \(\varphi^4_3\) measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to all coupling constants in any finite volume, and uniformly in the volume in the entire high temperature phases of the \(\varphi^4_2\) and \(\varphi^4_3\) models. The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron-Frobenius theorem, and bounds on the susceptibilities of the \(\varphi^4_2\) and \(\varphi^4_3\) measures obtained using skeleton inequalities.
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.
MSC:
| 81P05 | General and philosophical questions in quantum theory |
| 26E70 | Real analysis on time scales or measure chains |
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 74F05 | Thermal effects in solid mechanics |
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