Robust reconstruction on trees is determined by the second eigenvalue

Annals of Probability

Volumn 32, Issue 3 B, 2004, Pages 2630-2649

  Janson, Svante ^a   Mossel, Elchanan ^b  

^a UPPSALA UNIVERSITY   (Sweden)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Branching number; Reconstruction on trees; Robust phase transition

Indexed keywords

EID: 4544264380     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000153     Document Type: Article

References (29)

1. BERGER, N., KENYON, C., MOSSEL, E. and PERES, Y. (2004). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields. To appear.
2. BLEHER, P. M., RUIZ, J. and ZAGREBNOV, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473-482.
3. EVANS, W. S., KENYON, C., PERES, Y. and SCHULMAN, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410-433.
4. EVANS, W. S. and SCHULMAN, L. J. (1999). Signal propagation and noisy circuits. IEEE Trans. Inform. Theory 45 2367-2373.
5. FELSENSTEIN, J. (2003). Inferring Phylogenies. Sinauer, New York.
6. FURSTENBERG, H. (1970). Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (R. C. Gunning, ed.) 41-59. Princeton Univ. Press.
7. GEORGII, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
8. HAJEK, B. and WELLER, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388-291.
9. HIGUCHI, Y. (1977). Remarks on the limiting Gibbs states on a (d + 1)-tree. Publ. Res. Inst. Math. Sci. 13 335-348.
10. IOFFE, D. (1996). On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137-143.
11. KENYON, C., MOSSEL, E. and PERES, Y. (2001). Glauber dynamics on trees and hyperbolic graphs. In 42nd IEEE Symposium on on Foundations of Computer Science 568-578. IEEE, Los Alamitos, CA.
12. KESTEN, H. and STIGUM, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.
13. LYONS, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-353.
14. LYONS, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
15. MARTIN, J. (2003). Reconstruction thresholds on regular trees. In Discrete Random Walks (C. Banderier and C. Krattenthaler, eds.) 191-204. Available at http://dmtcs.loria.fr/proceedings/dmACind.html.
16. MARTINELLI, F., SINCLAIR, A. and WEITZ, D. (2003). The Ising model on trees: Boundary conditions and mixing time. In Proceedings of the Forty Fourth Annual Symposium on Foundations of Computer Science 628-639.
17. MARTINELLI, F., SINCLAIR, A. and WEITZ, D. (2004). The Ising model on trees: Boundary conditions and mixing time. Comm. Math. Phys. To appear.
18. MOSSEL, E. (2001). Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11 285-300.
19. MOSSEL, E. (2003). On the impossibility of reconstructing ancestral data and phylogenies. J. Comput. Biol. 10 669-678.
20. MOSSEL, E. (2004). Phase transitions in phylogeny. Trans. Amer. Math. Soc. 356 2379-2404.
21. MOSSEL, E. (2004). Survey: Information flow on trees. In Graphs, Morphisms and Statistical Physiscs (J. Nesetril and P. Winkler, eds.) 155-170. Amer. Math. Soc., Providence, RI.
22. MOSSEL, E. and PERES, Y. (2003). Information flow on trees. Ann. Appl. Probab. 13 817-844.
23. MOSSEL, E. and STEEL, M. (2004). A phase transition for a random cluster model on phylogenetic trees. Mathematical Biosciences 187 189-203.
24. PEMANTALE, R. and PERES, Y. (1995). Recursions on trees and the Ising model at critical temperatures. Unpublished manuscript.
25. PEMANTLE, R. and STEIF, J. E. (1999). Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 876-912.
26. SEMPLE, C. and STEEL, M. (2003). Phylogenetics. Oxford Univ. Press.
27. SOBER, E. and STEEL, M. A. (2002). Testing the hypothesis of common ancestry. J. Theor. Biol. 218 395-408.
28. SPITZER, F. (1975). Markov random fields on an infinite tree. Ann. Probab. 3 387-398.
29. VON NEUMANN, J. (1956). Probabilistic logics and the synthesis of reliable organisms from unreliable components. In Automata Studies (C. E. Shannon and J. McCarthy, eds.) 43-98. Princeton Univ. Press.