Phase transition for weakly interacting focusing Gibbs measures with harmonic potential
arXiv:2607.11608
The paper analyzes Gibbs measures for the focusing nonlinear Schrödinger equation with a harmonic potential as the interaction strength vanishes, identifying a critical threshold that separates convergence to a Gaussian measure from non‑convergence.
Abstract
In this paper, we study the Gibbs measures on Euclidean spaces associated to the focusing nonlinear Schrödinger equation with harmonic potential and critical non linearity whose coupling constant tends to 0, a question initially posed by Brydges-Slade (1996) for the -model on . In dimension one and in the higher dimensional cases (with radial assumption), we establish a critical threshold below which the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized cut-off) while, in the supercritical regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence.