Median and mean of the supremum of L² normalized random holomorphic fields

We prove that the expected value and median of the supremum of L² normalized random holomorphic fields of degree n on m-dimensional Kähler manifolds are asymptotically of order mlogn. There is an exponential concentration of measure of the sup norm around this median value. Prior results only gave the upper bound. The estimates are based on the entropy methods of Dudley and Sudakov combined with a precise analysis of the relevant distance functions and covering numbers using off-diagonal asymptotics of Bergman-Szegö kernels. Recent work on the critical value distribution is also used.