Moduli of rational curves in toric varieties and non-Archimedean geometry

We study the geometry of the Berkovich skeleton of the space of genus 0 logarithmic stable maps to a toric variety and give a canonical identification of this skeleton with a space of tropical maps. This result has two main applications. First, we use tropical geometry to give a compactification of the space of rational curves in a toric variety with prescribed contact order with the toric boundary. This compactification is then identified with the space of logarithmic stable maps. Second, we use these results to give a simple proof of the Nishinou--Siebert correspondence theorem for tropical curve counts without using toric degenerations. These results generalize work of Cavalieri, Markwig, and the author (arXiv:1410.2837) for target P^1 to arbitrary toric targets.