Directed polymer in random potentials on 5+1 and 6+1 dimensions

Abstract

Directed polymers in random potentials on higher dimensions at zero temperature are studied. The standard deviation Delta E(t) of the lowest energy E(t) of the polymer varies as t(beta) for length t and follows Delta E(L) similar to L-alpha at saturation, where L is the system size. We obtain beta = 0.130 +/- 0.006 and alpha = 0.232 +/- 0.007 in d = 5 + 1 and beta = 0.110 +/- 0.007 and alpha = 0.197 +/- 0.008 in d = 6 + 1. We measure the end to end distance Delta X (t) of the polymer and determine the dynamic exponent z directly by using the relation Delta X (t) similar to t(1/z). It is consistent with the values estimated from z = alpha/beta. They satisfy the scaling relation z + alpha = 2 very well. Our numerical results support that the upper critical dimension of the Kardar-Parisi-Zhang equation should be higher than d = 6 + 1. We also monitor the skewness and kurtosis of the energy distribution and find that they are in good agreement with the results for the discrete restricted solid-on-solid model. It seems that the normalized energy distribution is universal even in higher dimensions.