## Abstract

A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on *n* leaves, sample sequences of length Ω(log *n*) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than \({p^{\ast} = (\sqrt{2}-1)/2^{3/2}}\), then the tree can be recovered from sequences of length *O*(log *n*). The value *p** is given by 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*
^{Ω(1)}. Here we show that Steel’s conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from *O*(log *n*)-length sequences when the mutation probabilities are discretized and less than *p**. 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.

## References

- 1
Borgs, C., Chayes, J.T., Mossel, E., Roch, S.: The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In: FOCS, pp. 518–530 (2006)

- 2
Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Rel.

**131**(3), 311–340 (2005). In: Kenyon, C., Mossel, E., Peres, Y. (eds.) Proceedings of 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 568–578 (2001) - 3
Bleher P.M., Ruiz J., Zagrebnov V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys.

**79**(1–2), 473–482 (1995) - 4
Buneman, P.: The recovery of trees from measures of dissimilarity. In: Mathematics in the Archaelogical and Historical Sciences, pp. 187–395. Edinburgh University Press, Edinburgh (1971)

- 5
Cavender, J.A.: Taxonomy with confidence. Math. Biosci.

**40**(3–4) (1978) - 6
Chang J.T.: Full reconstruction of Markov models on evolutionary trees: identifiability and consistency. Math. Biosci.

**137**(1), 51–73 (1996) - 7
Daskalakis, C., Mossel, E., Roch, S.: Optimal phylogenetic reconstruction. In: STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 159–168. ACM, New York (2006)

- 8
Evans W.S., Kenyon C., Peres Y., Schulman L.J.: Broadcasting on trees and the Ising model. Ann. Appl. Probab.

**10**(2), 410–433 (2000) - 9
Erdös P.L., Steel M.A., Székely L.A., Warnow T.A.: A few logs suffice to build (almost) all trees (part 1). Random Struct. Algor.

**14**(2), 153–184 (1999) - 10
Farris J.S.: A probability model for inferring evolutionary trees. Syst. Zool.

**22**(4), 250–256 (1973) - 11
Felsenstein J.: Inferring Phylogenies. Sinauer, New York (2004)

- 12
Georgii H.O.: Gibbs measures and phase transitions, volume 9 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (1988)

- 13
Higuchi Y.: Remarks on the limiting Gibbs states on a (

*d*+ 1)-tree. Publ. Res. Inst. Math. Sci.**13**(2), 335–348 (1977) - 14
Ioffe D.: On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys.

**37**(2), 137–143 (1996) - 15
Jukes T.H., Cantor C.: Mammalian protein metabolism. In: Munro, H.N. (eds) Evolution of Protein Molecules, pp. 21–132. Academic, London (1969)

- 16
Janson S., Mossel E.: Robust reconstruction on trees is determined by the second eigenvalue. Ann. Probab.

**32**, 2630–2649 (2004) - 17
Kesten H., Stigum B.P.: Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Stat.

**37**, 1463–1481 (1966) - 18
Lyons R.: The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys.

**125**(2), 337–353 (1989) - 19
Mézard M., Montanari A.: Reconstruction on trees and spin glass transition. J. Stat. Phys.

**124**(6), 1317–1350 (2006) - 20
Mossel E.: Recursive reconstruction on periodic trees. Random Struct. Algor.

**13**(1), 81–97 (1998) - 21
Mossel E.: Reconstruction on trees: beating the second eigenvalue. Ann. Appl. Probab.

**11**(1), 285–300 (2001) - 22
Mossel E.: Phase transitions in phylogeny. Trans. Am. Math. Soc.

**356**(6), 2379–2404 (2004) - 23
Mossel E.: Distorted metrics on trees and phylogenetic forests. IEEE/ACM Trans. Comput. Bio. Bioinform.

**4**(1), 108–116 (2007) - 24
Mossel E., Peres Y.: Information flow on trees. Ann. Appl. Probab.

**13**(3), 817–844 (2003) - 25
Mossel E., Steel M.: A phase transition for a random cluster model on phylogenetic trees. Math. Biosci.

**187**(2), 189–203 (2004) - 26
Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, colorings, and other models on trees. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, pp. 449–458, 2004

- 27
Martinelli F., Sinclair A., Weitz D.: Glauber dynamics on trees: boundary conditions and mixing time. Comm. Math. Phys.

**250**(2), 301–334 (2004) - 28
Neyman J.: Molecular studies of evolution: a source of novel statistical problems. In: Gupta, S.S., Yackel, J. (eds) Statistical Desicion Theory and Related Topics, pp. 1–27. Academic Press, New York (1971)

- 29
Roch, S.: Sequence-length requirement for distance-based phylogeny reconstruction: breaking the polynomial barrier. In: FOCS, pp. 729–738 (2008)

- 30
Roch, S.: Phase transition in distance-based phylogeny reconstruction. Preprint (2009)

- 31
Sly, A.: Reconstruction of symmetric Potts models. Preprint available at http://arxiv.org/abs/0811.1208 (2008)

- 32
Spitzer F.: Markov random fields on an infinite tree. Ann. Probab.

**3**(3), 387–398 (1975) - 33
Semple C., Steel M.: Phylogenetics. Mathematics and its Applications Series, vol. 22. Oxford University Press, NY (2003)

- 34
Steel, M.: My Favourite Conjecture. Preprint (2001)

## Acknowledgments

S.R. thanks Martin Nowak and the Program for Evolutionary Dynamics at Harvard University where part of this work was done. E.M. and S.R. thank Mike Steel for his enthusiastic encouragement for studying the connections between the reconstruction problem and phylogenetics. We thank Satish Rao and Allan Sly for interesting discussions. We also thank the reviewers for their helpful comments. C.D. and S.R. performed this work at UC Berkeley and Microsoft Research.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## Author information

## Additional information

Partially supported by CIPRES (NSF ITR grant # NSF EF 03-31494). Supported by a Miller fellowship in Statistics and Computer Science, by a Sloan fellowship in Mathematics and by NSF grants DMS-0504245, DMS-0528488 DMS-0548249 (CAREER), ONR grant N0014-07-1-05-06 and ISF grant 1300/08.

## Rights and permissions

**Open Access** This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## About this article

### Cite this article

Daskalakis, C., Mossel, E. & Roch, S. Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture.
*Probab. Theory Relat. Fields* **149, **149–189 (2011). https://doi.org/10.1007/s00440-009-0246-2

Received:

Revised:

Published:

Issue Date:

### Keywords

- Phylogenetics
- CFN model
- Ising model
- Phase transitions
- Reconstruction problem
- Jukes–Cantor

### Mathematics Subject Classification (2000)

- Primary 60K35
- 92D15
- Secondary 60J85
- 82B26