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We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean-field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a ε=d_c-d expansion below the upper critical dimension d_c of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more ge
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