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We study the large scale equilibrium behavior of Van der Waals-Cahn-Hilliardphase transitions in stationary ergodic media. Specifically, we are interestedin free energy functionals of the following form \begin{equation*}\mathcal{F}^{\omega}(u) = \int_{\mathbb{R}^{d}} \left(\frac{1}{2}\varphi^{\omega}(x,Du(x))^{2} + W(u(x)) \right) \, dx, \end{equation*} where$W$ is a double-well potential and $\varphi^{\omega}(x,\cdot)$ is a stationaryergodic Finsler metric. We show that, at large scales, the random energy$\mathcal{F}^{\omega}$ can be approximated by the anisotropic perimeterassociated with a deterministic Finsler norm $\tilde{\varphi}$. To find$\tilde{\varphi}$, we build on existing work of Alberti, Bellettini, andPresutti, showing, in particular, that there is a natural sub-additive quantityin this context.

Surface Tension and $\Gamma$-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media