Large deviation principles for the Gross Pitaevskii Gibbs measure at low temperature

We prove the large deviation principle for the conditional Gibbs measure associated with the focusing Gross Pitaevskii equation in the low temperature regime. This conditional measure is of mixed type, being canonical in energy and microcanonical in particle number. In particular, our result extends the large deviation principle for the mixed ensemble studied by Ellis, Jordan, Otto, and Turkington to a more singular setting, where the interaction potential is unbounded and the conditional event involves diverging renormalization constants. As a consequence of the large deviation principle, the Gibbs measure concentrates along the soliton manifold in the low temperature limit.