  We compute the leading asymptotics of the maximum of the (centered) logarithm
of the absolute value of the characteristic polynomial, denoted $\Psi_N$, of
the Ginibre ensemble as the dimension $N$ of the random matrix tends to
infinity. The method relies on the log-correlated structure of the field
$\Psi_N$ and we obtain the lower-bound for the maximum by constructing a family
of Gaussian multiplicative chaos measures associated with certain
regularization of $\Psi_N$ at small mesoscopic scales. We also obtain the
leading asymptotics for the dimensions of the sets of thick points and verify
that they are consistent with the predictions coming from the Gaussian Free
Field. A key technical input is the approach from Ameur-Hedenmalm-Makarov to
derive the necessary asymptotics, as well as the results from Webb-Wong.
