Large scale regularity and correlation length for almost length-minimizing random curves in the plane

TL;DR

This paper investigates the large-scale regularity and correlation length of almost length-minimizing random curves in the plane, motivated by the RFIM and the Imry--Ma phenomenon, providing new bounds on phase boundary flatness.

Contribution

It introduces a continuum model for minimal surfaces under random fields, establishing refined estimates on the scale of boundary flatness influenced by quenched disorder.

Findings

Proves flatness of phase boundaries up to exponential scales in noise amplitude.

Connects the decay of correlations in RFIM to geometric properties of minimal surfaces.

Provides bounds on the influence of boundary conditions in the presence of quenched randomness.

Abstract

We consider a model of random curves in the plane related to the large-scale behavior of the Random Field Ising Model (RFIM) at temperature zero in two space dimensions. Our work is motivated by attempts to quantify the Imry--Ma phenomenon concerning the rounding of the phase transition by quenched disorder, and connects to recent advances regarding the decay of correlations in the RFIM. We study a continuum model of minimal surfaces in two space dimensions subject to an external, quenched random field, and restrict ourselves to isotropic surface integrands. The random fields we consider behave like white noise on large scales with an ultra-violet regularization reminiscent of the lattice structure of the RFIM. We give a finer description of the minimizer below the length scale starting from which the influence of boundary conditions is suppressed with a given probability, which has recently been shown to be of order $\exp(\varepsilon^{-\frac{4}{3}})$ in the amplitude $\varepsilon>0$ of the noise. More precisely, we prove flatness of the phase boundaries on scales up to order $\exp(\varepsilon^{-\frac{4}{13}})$.