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We study an optimal mass threshold for normalizability of the Gibbs measuresassociated with the focusing mass-critical nonlinear Schr\"odinger equation onthe one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer(1988) proposed a critical mass threshold given by the mass of the ground stateon the real line. We provide a proof for the optimality of this critical massthreshold. The proof also applies to the two-dimensional radial problem posedon the unit disc. In this case, we answer a question posed by Bourgain andBulut (2014) on the optimal mass threshold.  Furthermore, in the one-dimensional case, we show that the Gibbs measure isindeed normalizable at the optimal mass threshold, thus answering an openquestion posed by Lebowitz, Rose, and Speer (1988). This normalizability at theoptimal mass threshold is rather striking in view of the minimal mass blowupsolution for the focusing quintic nonlinear Schr\"odinger equation on theone-dimensional torus.

Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus