The (higher) q,t-Catalan numbers are a family of polynomials refining the sequence of (higher) Catalan numbers. These polynomials enjoy deep connections to the theory of symmetric functions, representation theory, algebraic geometry, and combinatorics. We give a combinatorial formula for the limiting distribution of coefficients of these polynomials analogous to the formula for q,t-Catalan numbers as a weighted sum over Dyck paths. In algebraic geometry language, the limiting distribution is the Duistermaat-Heckman measure of the punctual Hilbert scheme of points in the plane.