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

We consider the standard first passage percolation model on $$ with a distribution G taking two values $$ . 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.