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

Abstract

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 flow through the cylinder \([0,n]^ {d-1}\times [0,hn]\) 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 \(O(\frac{n^{d-1}}{\log n})\), for \(h\ge h_0\) (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 \([0,n]^ {d-1}\times [0,hn]\) is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant \(h\ge h_0\) (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.