Zero-temperature directed polymers in a random potential

Abstract

Extensive simulations of a model of directed polymers in a random potential are described. The standard deviation \ensuremath{\Delta}E(t) of the lowest energy E of walks of t steps varies as ${\mathit{t}}^{\mathrm{\ensuremath{\omega}}}$, but there is a large spread in the values quoted in the literature for \ensuremath{\omega} in d=2+1 and 3+1. In our model we are able to vary a parameter---the bending energy of the polymer---that affects the apparent value of \ensuremath{\omega}, but, if allowance is made for corrections to scaling, then we find that \ensuremath{\omega} is universal with the value 0.248\ifmmode\pm\else\textpm\fi{}0.004 in d=2+1 and \ensuremath{\omega}=0.20\ifmmode\pm\else\textpm\fi{}0.01 in d=3+1. The probability distribution P(E,t) is found to scale as t\ensuremath{\rightarrow}\ensuremath{\infty},P(E,t) dE\ensuremath{\rightarrow}P(a) da, where a=(E-〈E〉)/\ensuremath{\Delta}E. The scaling function P(a) is investigated and found to be universal, i.e., independent of such details as choice of distribution of the random potential, etc., and in fact identical to within our numerical accuracy to the equivalent distribution of heights in a model of ballistic aggregation.