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arXiv · 2310.08783

Optimal divergence rate of the focusing Gibbs measures

Damiano Greco·Guopeng Li·Rui Liang·Tadahiro Oh·Yuzhao Wang
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Abstract

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).

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Damiano Greco, Guopeng Li, Rui Liang, Tadahiro Oh, Yuzhao Wang. 2023-10-13. Optimal divergence rate of the focusing Gibbs measures. https://arxiv.org/abs/2310.08783

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