Phase transitions in phylogeny

We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies.

We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least $3$, and the net transition on each edge is bounded by $\epsilon$. Motivated by a conjecture by M. Steel, we show that if $2 (1 - 2 \epsilon)^2 > 1$, then for balanced trees, the topology of the underlying tree, having $n$ leaves, can be reconstructed from $O(\log n)$ samples (characters) at the leaves. On the other hand, we show that if $2 (1 - 2 \epsilon)^2 < 1$, then there exist topologies which require at least $n^{\Omega(1)}$ samples for reconstruction.

Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.