Local limit of focusing discrete Non linear Schrodinger equation

We examine the behavior of a function sampled from the invariant measure asso- ciated to the focusing discrete Non Linear Schrodinger equation, defined on a discrete torus of dimension d≥3, and nonlinearity parameter p > 4, in the infinite volume limit. This is the same as a Gaussian free field conditioned to have l2 mass O(volume of the torus), exponentially tilted by lp norm. The Gibbs measure has two parameters, the inverse temperature and the strength of the non linearity, and the scaling is such that the non linear and linear parts of the Hamiltonian contribute on the same scale. We prove that there are three regions of the phase diagram which yield three distinct local limits, a massive Gaussian free field, a massless Gaussian free field plus a random constant, and finally a (possibly trivial) mixture of massive Gaussian free fields, where some mass is “lost” to the region of concentration. Our proof relies on the analysis of the spherical model of a Ferromagnet and a delicate removal of the `soliton' part of the measure so that the model reduces to the spherical model locally. Joint work with Kesav Krishnan.