Abstract
We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.
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Derrida, B., Spohn, H. Polymers on disordered trees, spin glasses, and traveling waves. J Stat Phys 51, 817–840 (1988). https://doi.org/10.1007/BF01014886
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DOI: https://doi.org/10.1007/BF01014886