Numerical Results for the Ground-State Interface in a Random Medium

TL;DR

This paper uses a minimum-cut algorithm to numerically determine the ground state of a $d$-dimensional interface in a random medium, providing precise estimates of roughness and energy exponents in 2D and 3D.

Contribution

It introduces a numerical method that accurately finds ground states for complex interface problems, improving upon previous inexact and slow techniques.

Findings

Roughness exponent in 2D: 0.41 ± 0.01

Roughness exponent in 3D: 0.22 ± 0.01

Energy exponent in 2D: 0.84 ± 0.03

Abstract

The problem of determining the ground state of a $d$-dimensional interface embedded in a $(d+1)$-dimensional random medium is treated numerically. Using a minimum-cut algorithm, the exact ground states can be found for a number of problems for which other numerical methods are inexact and slow. In particular, results are presented for the roughness exponents and ground-state energy fluctuations in a random bond Ising model. It is found that the roughness exponent $\zeta = 0.41 \pm 0.01, 0.22 \pm 0.01$, with the related energy exponent being $\theta = 0.84 \pm 0.03, 1.45 \pm 0.04$, in $d = 2, 3$, respectively. These results are compared with previous analytical and numerical estimates.