  In order to probe with high precision the tails of the ground-state energy
distribution of disordered spin systems, K\"orner, 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
$\eta(d)$ 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 simple
formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the
similarities and differences with spin-glasses.
