  The Ising model is one of the simplest and most well-established concepts to
simulate phase transformations in complex materials. However, its most central
constant, the interaction strength J between two nearest neighbors, is hard to
obtain. Here we show how this basic constant can be determined with a fractal
dimension analysis of measured domain structures. We apply this approach to
vanadium dioxide, a strongly correlated material with a first-order
insulator-to-metal phase-transition with enigmatic properties. We obtain a
nearest-neighbor interaction of 13.8 meV, a value close to the thermal energy
at room temperature. Consequently, even far below the transition temperature,
there are spontaneous local phase-flips from the insulating into the metallic
phase. These fluctuations explain several measured anomalies in VO$_2$, in
particular the low thermal carrier activation energy and the finite
conductivity of the insulating phase. As a method, our fractal dimension
analysis links the Ising model to macroscopic material constants for almost any
first-order phase transition.

Statistical Mechanics;Strongly Correlated Electrons