  In this paper we study the distribution of successive minima of global
sections of powers of a metrized ample line bundle on a variety over a number
field. We develop criteria for there to exist a measure on the real line
describing the limiting behavior of this distribution as one considers
increasing powers of the bundle. When this measure exists, we develop methods
for determining it explicitly. We present applications to the distribution of
Petersson norms of cusp forms of increasing weight for SL_2(Z) and to the
minimal sup norm of algebraic functions on adelic subsets of curves arising in
capacity theory.
