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

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