Tournament limits: Degree distributions, score functions and self-converseness

Motivated by known results for finite tournaments, we define and study the score functions of tournament kernels and the degree distributions of tournament limits. Our main theorem completely characterises those distributions that appear as the degree distribution of some tournament limit and those functions that appear as the score function of some tournament kernel. We also show that only the uniform distribution can be realised as the outdegree distribution of a unique tournament limit. Finally we define self-converse tournament limits and kernels and characterise their degree distributions and score functions.