Journal of Statistical Physics, Vol. 79. Nos. 1/2, 1995. p. 117-164.
We investigate the behavior of various spin-systems that are subject to the highly correlated and extremely diluted quenched disorder as provided by the fractal aerogel model. For these systems, it is (easily) established that, at all temperatures, the free energy is identical to that of the corresponding uniform system. The surface tension, however, behaves quite differently. Foremost, at any fixed temperature corresponding to the low temperature phase in the uniform system, there is a non-trivial curve in the aerogel phase plane dividing
high-temperature behavior (zero surface tension) from low-temperature behavior (positive surface tension). The fractal aerogel has two distinctive phases in its own right: gel and sol. In the gel phase, the spin system has zero surface tension at all temperatures. In one region of the sol phase, the surface tension is shown to be equal to its value in the uniform system. Since part of this region borders on the gel phase, a certain portion of the sol/gel phase boundary is also the dividing line between high- and low-temperature behavior. Evidently, in this case, the surface tension is discontinuous at the phase boundary, on the other hand, there are well-defined length scales that diverge as the phase boundary is approached. This demonstrates an absence of scaling in these systems.