Exact enumeration approach to the directed polymer problem

An exact enumeration approach is developed for the directed polymer problem. The probability distribution of the number of directed self-avoiding walks that can reach a certain level t is obtained exactly up to t=10. This enables us to calculate some properties of directed polymers that are not attainable by Monte Carlo simulations. Specifically, we find that the fluctuation of the logarithm of the number of directed self-avoiding walks that can reach level t, when averaged over the configurations that can reach level t, scales as ${\mathit{t}}^{1/2}$ well below the directed percolation threshold ${\mathit{p}}_{\mathit{c}\mathit{D}}$, contrary to the behavior ${\mathit{t}}^{1/3}$, which is known to be valid when the bond probability p is above ${\mathit{p}}_{\mathit{c}\mathit{D}}$. When p is close to 1, these fluctuations scale as ${\mathit{t}}^{1/5}$ for a very long time before the true asymptotic behavior ${\mathit{t}}^{1/3}$ is recovered. The method can also be used to obtain the behavior of averages of moments of the number of directed self-avoiding walks that can reach level t. Below ${\mathit{p}}_{\mathit{c}\mathit{D}}$ these quantities are dominated by rare configurations and cannot be obtained by Monte Carlo simulations.