Optimal divergence rate of the focusing Gibbs measures
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908512806764544 |
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| author | Greco, Damiano Li, Guopeng Liang, Rui Oh, Tadahiro Wang, Yuzhao |
| author_facet | Greco, Damiano Li, Guopeng Liang, Rui Oh, Tadahiro Wang, Yuzhao |
| contents | We study Gibbs measures on the $d$-dimensional torus with $L^2$-(super)critical focusing interaction potentials. We establish a precise divergence rate of the partition function as we remove regularization, where the optimal constant is given by (i) (the negative of) the minimum value of the Hamiltonian given an $L^2$-constraint in the $L^2$-critical case and (ii) the optimal constant for certain Bernstein's inequality in the mass-supercritical case. In particular, our result in the $L^2$-critical case precisely quantifies the phase transition of the focusing Gibbs measure at the critical $L^2$ threshold, previously studied by Lebowitz, Rose, and Speer (1988) and Sosoe, Tolomeo, and the fourth author (2022). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08783 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal divergence rate of the focusing Gibbs measures Greco, Damiano Li, Guopeng Liang, Rui Oh, Tadahiro Wang, Yuzhao Probability Mathematical Physics Analysis of PDEs 60H30, 81T08, 35Q55, 60H40 We study Gibbs measures on the $d$-dimensional torus with $L^2$-(super)critical focusing interaction potentials. We establish a precise divergence rate of the partition function as we remove regularization, where the optimal constant is given by (i) (the negative of) the minimum value of the Hamiltonian given an $L^2$-constraint in the $L^2$-critical case and (ii) the optimal constant for certain Bernstein's inequality in the mass-supercritical case. In particular, our result in the $L^2$-critical case precisely quantifies the phase transition of the focusing Gibbs measure at the critical $L^2$ threshold, previously studied by Lebowitz, Rose, and Speer (1988) and Sosoe, Tolomeo, and the fourth author (2022). |
| title | Optimal divergence rate of the focusing Gibbs measures |
| topic | Probability Mathematical Physics Analysis of PDEs 60H30, 81T08, 35Q55, 60H40 |
| url | https://arxiv.org/abs/2310.08783 |