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

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.