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We prove that the expected value and median of the supremum of $L^2$normalized random holomorphic fields of degree $n$ on $m$-dimensional K\"ahlermanifolds are asymptotically of order $\sqrt{m\log n}$. This improves the priorresult of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of themedia is of order $\sqrt{\log n}$  The estimates are based on the entropy methods of Dudley and Sudakov combinedwith a precise analysis of the relevant pseudo-metric and its covering numbers,which can be precisely evaluated using off-diagonal asymptotics of Bergmankernels. Recent work of the authors on the value distribution of these fieldsare also used to get precise constants.

Median and mean of the Supremum of $L^2$ normalized random holmorphic fields