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

Abstract

AbstractWe consider the standard first passage percolation model on $${\mathbb {Z}}^ d$$ Z d with a distribution G taking two values $$0<a<b$$ 0 < a < b . We study the maximal flow through the cylinder $$[0,n]^ {d-1}\times [0,hn]$$ [ 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 $$O(\frac{n^{d-1}}{\log n})$$ O ( n d - 1 log n ) , for $$h\ge h_0$$ h ≥ h 0 (for a large enough constant $$h_0=h_0(a,b)$$ h 0 = h 0 ( a , b ) ). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder $$[0,n]^ {d-1}\times [0,hn]$$ [ 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 $$h\ge h_0$$ 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.