First passage percolation with long-range correlations and applications to random Schrödinger operators

Abstract

We consider first passage percolation (FPP) with passage times generated by a general class of models with long-range correlations on ${\mathbb{Z}}^{\mathit{d}}$, $\mathit{d}\ge 2$, including discrete Gaussian free fields, Ginzburg–Landau $\nabla \mathit{\varphi }$ interface models or random interlacements as prominent examples. We show that the associated time constant is positive, the FPP distance is comparable to the Euclidean distance, and we obtain a shape theorem. We also present two applications for random conductance models (RCM) with possibly unbounded and strongly correlated conductances. Namely, we obtain a Gaussian heat kernel upper bound for RCMs with a general class of speed measures, and an exponential decay estimate for the Green’s function of RCMs with random killing measures.