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We consider the standard first passage percolation model on $\mathbb Z^ d$with a distribution $G$ taking two values $0<a<b$. We study the maximal flowthrough the cylinder $[0,n]^ {d-1}\times [0,hn]$ between its top and bottom aswell as its associated minimal surface(s). We prove that the variance of themaximal flow is superconcentrated, i.e. in $O(\frac {n^{d-1}} {\log n})$, for$h\geq h_0$ (for a large enough constant $h_0=h_0(a,b)$). Equivalently, weobtain that the ground state energy of a disordered Ising ferromagnet in acylinder $[0,n]^ {d-1}\times [0,hn]$ is superconcentrated when oppositeboundary conditions are applied at the top and bottom faces and for a largeenough constant $h\geq h_0$ (which depends on the law of the couplingconstants). Our proof is inspired by the proof of Benjamini--Kalai--Schramm.Yet, one major difficulty in this setting is to control the influence of theedges since the averaging trick used in the proof of Benjamini--Kalai--Schrammfails for surfaces. Of independent interest, we prove that minimal surfaces (inthe present discrete setting) cannot have long thin chimneys.

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