Directed paths on percolation clusters |
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Authors: | Leon Balents Mehran Kardar |
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Institution: | (1) Department of Physics, Harvard University, 02138 Cambridge, Massachusetts;(2) Department of Physics, Massachusetts Institute of Technology, 02139 Cambridge, Massachusetts |
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Abstract: | We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft
and xt
, describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold. |
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Keywords: | Directed polymers percolation random walks hierarchical lattices disorder |
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