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A Gaussian variational approximation is often used to study interfaces inrandom media. By considering the 1+1 dimensional directed polymer in a randommedium, it is shown here that the variational Ansatz typically leads to anegative variance of the free energy. The situation improves by taking intoaccount more and more steps of replica symmetry breaking. For infinite orderbreaking the variance is zero (i.e. subextensive).  This situation is reminiscent of the negative entropies in mean field spinglass models, which were also eliminated by considering infinite order replicasymmetry breaking.Comment: 8 pages revtex. Submitted to Journal de Physique (France

Variational Approach to Interfaces in Random Media: Negative Variances and Replica Symmetry Breaking