Random zero currents of sections of Hermitian line bundles over compact Riemannian manifolds

This paper is concerned with zero currents of random section of a Hermitian line bundle $E$ over a compact oriented Riemannian manifold. Given a metric connection, heat flow yields a natural 1-parameter family of probability measures on the space of smooth sections $\mathrm{\Gamma }E$. It is shown that the corresponding family of random zero currents connects the curvature of the bundle to the ground state zero current.