Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets.

We consider the standard first passage percolation model on with a distribution taking two values . We study the maximal flow through the cylinder between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in , for (for a large enough constant ). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini-Kalai-Schramm (Ann Probab 31:1970-1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970-1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.