Title: Phase transition in the Discrete Non Linear Schrodinger Equation.
Abstract: The Non Linear Schrodinger Equation is a canonical example of a dispersive PDE that can also display stable, spatially localized solutions called solitons. Invariant measures for the flow of the equation have been used to study not only the well-posedness of the equation, but also the typicality of the long term behavior, whether dispersive or solitonic. In this talk, I will review some of the existing results and then describe joint work with Partha Dey and Kay Kirkpatrick on the corresponding Discrete PDE in dimension 3 and higher. In particular, I will define a family of invariant Gibbs measures for the discrete equations where the key parameter is the strength of the non linearity. We prove converence of the associated free enery, and as the strength of the non linearity is varied, we establish existence of a phase transition. In the supercritical regime the support of the measure lies on very sharply peaked functions corresponding to a soliton phase, and resembles the Gaussian free field conditioned to have given $\ell^{2}$ norm in the subcritical regime.
