Minimal surfaces in a random environment arise naturally in the homogenization of models in fracture mechanics. They also are one of the natural higher-dimensional generalizations of planar first-passage percolation, the problem of finding the shortest path in a random environment. Under suitable assumptions on the probability distribution - assuming in particular statistical isotropy of the random medium -, we establish an algebraic rate of convergence for the energy of the minimal surface dividing a box $(-R,R)^d$ in the limit $R\rightarrow \infty$. We also prove a superconcentration principle for energy fluctuations in up to four ambient spatial dimensions: Energy fluctuations of the minimal surface are suppressed, as compared to the the energy fluctuations of a fixed surface like $(-R,R)^{d-1} \times \{0\}$. We discuss possible implications of the superconcentration principle concerning the accuracy of variational models for fracture in the setting of heterogeneous media. joint works with Antonio Agresti, Nicolas Clozeau, Assylbek Olzhabayev, Christian Wagner