Upper and Lower Bounds for the Correlation Length of the Two-Dimensional Random-Field Ising Model

TL;DR

This paper establishes upper and lower bounds on the correlation length in the 2D random-field Ising model at weak fields, revealing exponential decay and persistent boundary dependence at large scales.

Contribution

It provides the first explicit exponential upper bound on correlation length and demonstrates boundary condition influence on large domains, advancing understanding of phase behavior.

Findings

Correlation length bounded by double exponential in 1/ε^2

Strong boundary dependence persists on large domains

Exponential decay of correlations at weak fields

Abstract

We study the rate of correlation decay in the two-dimensional random-field Ising model at weak field strength $\varepsilon$. We combine elements of the recent proof of exponential decay of correlations with a quantitative refinement of a result of Aizenman--Burchard on the tortuosity of random curves to obtain an upper bound of the form $\exp(\exp(O(1/\varepsilon^{2})))$ on the correlation length of the model at all temperatures. Conversely, we show, by adapting methods of Fisher--Fr\"{o}hlich--Spencer, that on square domains of side length as large as $\exp(O(1/\varepsilon^{2/3}))$ the model continues to exhibit strong dependence on boundary conditions at low temperature.