The interplay between theta and T

We extend a recent computation of the dependence of the free energy, F, on the non-commutative scale theta to theories with very different UV sensitiv ity. The temperature dependence of F strongly suggests that a reduced numbe r of degrees of freedom contributes to the free energy in the non-planar se ctor, F-np, at high temperature. This phenomenon seems generic, independent of the UV sensitivity, and can be traced to modes whose thermal wavelength s become smaller than the noncommutativity scale. The temperature dependenc e of F-np can then be calculated at high temperature using classical statis tical mechanics, without encountering a UV catastrophe even in large number of dimensions. This result is a telltale sign of the low number of degrees of freedom contributing to F in the non-planar sector at high temperature. Such behavior is in marked contrast to what would happen in a field theory with a random set of higher derivative interactions.