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We analyze the thermodynamics of the focusing discrete nonlinearSchr\"odinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$and under a model with two parameters, representing inverse temperature andstrength of the nonlinearity, respectively. We prove the existence of limitingfree energy and analyze the phase diagram for general $d,p$. We also prove theexistence of a continuous phase transition curve that divides the parametricplane into two regions involving the appearance or non-appearance of solitons.Appropriate upper and lower bounds for the curve are constructed. We also lookat the typical behavior of a function chosen from the Gibbs measure for certainparts of the phase diagram.

Phase Transition for Discrete Non Linear Schr\"odinger Equation in Three and Higher Dimensions