Probability of phase separation for the Bose gas with delta interactions

We consider the quantum Non-linear Schr\"{o}dinger equation $i\partial_t\Psi=-\partial_x^2\Psi +2c\Psi^\dagger\Psi^2$ with positive coupling constant $c$ varying from zero to infinity. We study quantum correlation functions of this model using the determinant representation of these correlation functions. We consider the case of a finite density ground state and evaluate, using the Riemann Hilbert problem, the asymptotics of the probability that no particles are present in the space interval $[0,x]$ in the large $x$ limit. We call this the probability of phase separation or alternately the emptiness formation probability.