We calculate, by performing a product of random matrices, the specific heat and its derivative for the problem of directed polymers in a random medium. Our results are consistent with the existence of a phase transition both in 3+1 and 2+1 dimensions and with the absence of a phase transition in 1+1 dimensions. Our finite-size scaling analysis leads to {α}=-0.1±{}0.1 in 3+1 and {α}=-1.0±{}0.2 in 2+1 dimensions.
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Derrida et al. (1990) studied this question.
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