Curves with a rational MLE and chipfiring games

This talk concerns statistical models in the n-simplex where the Maximum Likelihood Estimator (MLE) is a rational function. Eliana Duarte, Orlando Marigliano and Bernd Sturmfels recently proved that such models all arise as the image of a so-called Horn map. In their paper, they ask whether models with a rational MLE can be classified. We study the case where the models have dimension 1. In this case, after some simple reduction steps, such models correspond to outcomes of directed chipfiring games (or actually, more conveniently chipsplitting games) on a certain graph. We conjecture an upperbound on the size of these outcomes, which gives an upperbound on the degree of the corresponding models, and prove this conjecture for n<=4. This is joint work with Orlando Marigliano.