  Motivated by recently discovered relations between logarithmically correlated
Gaussian processes and characteristic polynomials of large random $N \times N$
matrices $H$ from the Gaussian Unitary Ensemble (GUE), we consider the problem
of characterising the distribution of the global maximum of
$D_{N}(x):=-\log|\det(xI-H)|$ as $N \to \infty$ and $x\in (-1,1)$. We arrive at
an explicit expression for the asymptotic probability density of the
(appropriately shifted) maximum by combining the rigorous Fisher-Hartwig
asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing
transition} scenario for logarithmically correlated processes. Although the
general idea behind the method is the same as for the earlier considered case
of the Circular Unitary Ensemble, the present GUE case poses new challenges. In
particular we show how the conjectured {\it self-duality} in the freezing
scenario plays the crucial role in our selection of the form of the maximum
distribution. Finally, we demonstrate a good agreement of the found probability
density with the results of direct numerical simulations of the maxima of
$D_{N}(x)$.
