  We compute the leading asymptotics as $N\to\infty$ of the maximum of the
field $Q_N(q)= \log\det|q- A_N|$, $q\in \mathbb{C}$, for any unitarily
invariant Hermitian random matrix $A_N$ associated to a non-critical
real-analytic potential. Hence, we verify the leading order in a conjecture of
Fyodorov and Simm formulated for the GUE. The method relies on a classical
upper-bound and a more sophisticated lower-bound based on a variant of the
second-moment method which exploits the hyperbolic branching structure of the
field $Q_N(q)$, $q$ in the upper half plane. Specifically, we compare $Q_N$ to
an idealized Gaussian field by means of exponential moments. In principle, this
method could also be applied to random fields coming from other point processes
provided that one can compute certain mixed exponential moments. For unitarily
invariant ensembles, we show that these assumptions follow from the
Fyodorov-Strahov formula and asymptotics of orthogonal polynomials derived by
Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou.
