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

We consider the standard first passage percolation model on Z d with a distribution G taking two values 0 < a < b . We study the maximal flow through the cylinder [ 0 , n ] d - 1 × [ 0 , h n ] 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 O (n d - 1 log n) , for h ≥ h 0 (for a large enough constant h 0 = h 0 (a , b) ). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder [ 0 , n ] d - 1 × [ 0 , h n ] is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant h ≥ h 0 (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. [ABSTRACT FROM AUTHOR]