Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$

The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications. In the second part of the paper, we apply our formula to the moduli space $\overline{M}_{0, n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps $\overline{M}_{0, n}\to\mathbb{P}^{n-3}$, whose hyperplane pullbacks are the $\psi$-classes, with respect to a suitable choice of torus. We introduce tropical $\psi$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $\psi$-hypersurfaces.