  We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical
mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS)
equation to a more general class of models, including higher dimensionalities
and nonlinearities of arbitrary degree. These extensions are physically
motivated by the desire to describe situations with an excitation threshold for
creation of localized excitations, as well as by recent work suggesting
non-cubic DNLS models to describe Bose-Einstein condensates in deep optical
lattices, taking into account the effective condensate dimensionality.
Considering ensembles of initial conditions with given values of the two
conserved quantities, norm and Hamiltonian, we calculate analytically the
boundary of the 'normal' Gibbsian regime corresponding to infinite temperature,
and perform numerical simulations to illuminate the nature of the localization
dynamics outside this regime for various cases. Furthermore, we show
quantitatively how this DNLS localization transition manifests itself for
small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled
anharmonic oscillators (in which energy is the only conserved quantity), and
determine conditions for existence of persistent energy localization over large
time scales.
