We prove that the expected value and median of the supremum of L^2 normalized random holomorphic fields of degree n on m-dimensional Kahler manifolds are asymptotically of order \sqrt{m\log n}.

There is an exponential concentration of measure of the sup norm around this median value. 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 kernels. This is the joint work with S. Zelditch.