Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder

Abstract

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, Körner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo Markov chain in the disorder. In this paper, we combine their Monte-Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution $P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $η(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{η(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $θ(d)$ (which governs the fluctuations $ΔE_0(L) \sim L^{θ(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $η(d)=1/(1-θ(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.