Random Polynomials of High Degree and Levy Concentration of Measure

We show that the Lp norms of random sequences of holomorphic sections sN ∈ H(M, L ) of powers of a positive line bundle L over a compact Kähler manifold M satisfy ‖sN‖p/‖sN‖2 = { O(1) for 2 ≤ p < ∞ O( √ logN) for p = ∞ } almost surely. This estimate also holds for almost-holomorphic sections of positive line bundles on symplectic manifolds (in the sense of our previous work) and we give almost sure bounds for the Ck norms. Our methods involve asymptotics of Bergman-Szegö kernels and the concentration of measure phenomenon.