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The nonlinear Schrödinger equation,, arises from a Hamiltonian on infinite-dimensional phase space. For, Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measureon ballsin phase space such that the Cauchy problem foris well posed on the support of, and thatis invariant under the flow. This paper shows thatsatisfies a logarithmic Sobolev inequality (LSI) for the focusing caseandonfor allN>0; alsosatisfies a restricted LSI foron compact subsets ofdetermined by Hölder norms. Hence forp = 4, the spectral data of the periodic Dirac operator inwith random potentialsubject toare concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries. [ABSTRACT FROM PUBLISHER]