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Log-Sobolev inequality for the sinh-Gordon model

  • Omar Abdelghani
  • Roland Bauerschmidt
  • Thierry Bodineau
  • Benoit Dagallier
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Abstract

For the sinh-Gordon model (without external mass term), we show that the log-Sobolev inequality holds uniformly in the lattice and volume regularisations for all parameters in the regime in which the Gaussian multiplicative chaos has second moments. This implies that infinite-volume ``massless'' sinh-Gordon measures on ℝ2 satisfying all Osterwalder--Schrader axioms exist and that they have exponential decay of correlations (are ``massive''). The proof uses the Polchinski equation method for log-Sobolev inequalities. The required estimates on the renormalised potential are established using simple perturbation theoretic bounds, valid for absolutely monotone potentials, and an extension of the correlation inequality of Ding--Song--Sun to potentials in the GHS class. This inequality is proved also by using a variant of the Polchinski equation through the maximum principle.

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Cite

BibTeXRIS
BibTeX
@misc{abdelghani2026logsobolev,
  title = {{Log-Sobolev inequality for the sinh-Gordon model}},
  author = {Omar Abdelghani and Roland Bauerschmidt and Thierry Bodineau and Benoit Dagallier},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.28091},
  eprint = {2609.28091},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28091},
}
RIS
TY  - UNPB
TI  - Log-Sobolev inequality for the sinh-Gordon model
AU  - Omar Abdelghani
AU  - Roland Bauerschmidt
AU  - Thierry Bodineau
AU  - Benoit Dagallier
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.28091
UR  - https://arxiv.org/abs/2609.28091
AB  - For the sinh-Gordon model (without external mass term), we show that the log-Sobolev inequality holds uniformly in the lattice and volume regularisations for all parameters in the regime in which the Gaussian multiplicative chaos has second moments. This implies that infinite-volume ``massless'' sinh-Gordon measures on ℝ^2 satisfying all Osterwalder--Schrader axioms exist and that they have exponential decay of correlations (are ``massive''). The proof uses the Polchinski equation method for log-Sobolev inequalities. The required estimates on the renormalised potential are established using simple perturbation theoretic bounds, valid for absolutely monotone potentials, and an extension of the correlation inequality of Ding--Song--Sun to potentials in the GHS class. This inequality is proved also by using a variant of the Polchinski equation through the maximum principle.
ER  -