Optimal divergence rate of the focusing Gibbs measures

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).