  We study the statistical mechanics of the one-dimensional discrete nonlinear
Schr\"odinger (DNLS) equation with saturable nonlinearity. Our study represents
an extension of earlier work [Phys. Rev. Lett. {\bf 84}, 3740 (2000)] regarding
the statistical mechanics of the one-dimensional DNLS equation with a cubic
nonlinearity. As in this earlier study we identify the spontaneous creation of
localized excitations with a discontinuity in the partition function. The fact
that this phenomenon is retained in the saturable DNLS is non-trivial, since in
contrast to the cubic DNLS whose nonlinear character is enhanced as the
excitation amplitude increases, the saturable DNLS in fact becomes increasingly
linear as the excitation amplitude increases. We explore the nonlinear dynamics
of this phenomenon by direct numerical simulations.
