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

摘要

In order to probe with high precision the tails of the ground-state energydistribution of disordered spin systems, K\"orner, Katzgraber and Hartmann\cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-CarloMarkov chain in the disorder. In this paper, we combine their Monte-Carloprocedure in the disorder with exact transfer matrix calculations in eachsample 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 withthe exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and$d=3$, we measure the negative tail up to ten standard deviations, whichcorrespond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results arein agreement with Zhang's argument, stating that the negative tail exponent$\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$as $E_0 \to -\infty$ is directly related to the fluctuation exponent$\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$of the ground state energy $E_0$ for polymers of length $L$) via the simpleformula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on thesimilarities and differences with spin-glasses.