q,t-Catalan measures

We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on ${\mathbb{R}}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the $q,t$-Catalan numbers. Our main result realizes the $q,t$-Catalan measures as a limit of higher $q,t$-Catalan numbers $C_n^{(m)}(q,t)$ as $m\to \mathrm{\infty }$. We also give a geometric interpretation of the $q,t$-Catalan measures. They are the Duistermaat-Heckman measures of the punctual Hilbert schemes parametrizing subschemes of ${\mathbb{C}}^2$ supported at the origin.