Rich ecosystems harbour thousands of species interacting in tangled networks encompassing predation, mutualism and competition. Such widespread biodiversity is puzzling, because in ecological models it is exceedingly improbable for large communities to stably coexist. One aspect rarely considered in these models, however, is that coexisting species in natural communities are a selected portion of a much larger pool, which has been pruned by population dynamics. Here we compute the distribution of the number of species that can coexist when we start from a pool of species interacting randomly, and show that even in this case we can observe rich, stable communities. Interestingly, our results show that, once stability conditions are met, network structure has very little influence on the level of biodiversity attained. Our results identify the main drivers responsible for widespread coexistence in natural communities, providing a baseline for determining which structural aspects of empirical communities promote or hinder coexistence.

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We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice $A_1,\dots,A_k$ are randomly selected. We conjecture that the probability of ties goes to 0 as $n$ grows. We conjecture and provide some supporting evidence that---contrary to intuition---each of the $2^{k \choose 2}$ assignments of $\succ$ or $\prec$ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select $k$ dice $A_1,\dots,A_k$ and observe that $A_1 \succ A_2 \succ \ldots \succ A_k$. Then our conjecture asserts that the outcomes $A_k \succ A_1$ and $A_1 \prec A_k$ both have probability approaching $1/2$ as $n \rightarrow \infty$