Evolutionary Trees and the Ising Model on the Bethe Lattice: a Proof of Steel's Conjecture

One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than p∗ = ( √ 2 − 1)/2, then the tree can be recovered from sequences of length O(log n). The value p∗ is exactly the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel’s conjecture was proven by the second author in the special case where the tree is “balanced”. The second author also proved that if all edges have mutation probability larger than p∗ then the length needed is n. Here we complete the proof of Steel’s conjecture and give a reconstruction algorithm that requires optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain an optimal reconstruction algorithm for the Jukes-Cantor model with short edges. All reconstruction algorithms run in polynomial time. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.