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

TL;DR

This paper demonstrates superconcentration phenomena for maximal flow and minimal surfaces in first passage percolation and disordered Ising models, showing variance bounds and structural properties of minimal surfaces.

Contribution

It establishes superconcentration bounds for maximal flow and ground state energy in disordered systems, extending techniques inspired by Benjamini--Kalai--Schramm to surface settings.

Findings

Variance of maximal flow is O(n^{d-1}/log n) for large h

Minimal surfaces cannot have long thin chimneys

Superconcentration of ground state energy in 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 $h\geq h_0$ (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 $h\geq h_0$ (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini--Kalai--Schramm. Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in the proof of Benjamini--Kalai--Schramm fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.