Consider a Markov chain on an infinite tree T=(V,E) rooted at ρ. In such a chain, once the initial root state σ({ρ}) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let μj denote the resulting measure for σ({ρ})=j. The resulting measure μj is defined on configurations 
, where 
is some finite set. Let μjn denote the restriction of μ to the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting 
, where dTV denotes total variation distance, we say that the reconstruction problem is solvable if lim inf n→∞αn>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n→∞.
In this paper we study the problem of robust reconstruction. Let ν be a nondegenerate distribution on 
and ɛ>0. Let σ be chosen according to μjn and σ' be obtained from σ by letting for each node independently, σ(v)=σ'(v) with probability 1−ɛ and σ'(v) be an independent sample from ν otherwise. We denote by μjn[ν,ɛ] the resulting measure on σ'. The measure μjn[ν,ɛ] is a perturbation of the measure μjn. Letting ![$\alpha_{n}(\nu,\varepsilon )=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n}[\nu,\varepsilon ],\mu_{j}^{n}[\nu,\varepsilon ])$](/DPubS?service=Repository&version=1.0&verb=Disseminate&view=body&handle=euclid.aop/1091813626&content-type=gif_5)
, we say that the reconstruction problem is ν-robust-solvable if lim inf n→∞αn(ν,ɛ)>0 for all 0<ɛ<1. Roughly speaking, the reconstruction problem is robust-solvable if for any noise-rate and for all n, the nth level of the tree contains a nonvanishing amount of information on the root of the tree.
Standard techniques imply that if T is the rooted B-ary tree (where each node has B children) and if B|λ2(M)|2>1, where λ2(M) is the second largest eigenvalue of M (in absolute value), then for all nondegenerate ν, the reconstruction problem is ν-robust-solvable. We prove a converse and show that the reconstruction problem is not ν-robust-solvable if B|λ2(M)|2<1. This proves a conjecture by the second author and Y. Peres. We also consider other models of noise and general trees.
References
Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2004). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields. To appear.
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.
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.
Evans, W. S. and Schulman, L. J. (1999). Signal propagation and noisy circuits. IEEE Trans. Inform. Theory 45 2367--2373.
Felsenstein, J. (2003). Inferring Phylogenies. Sinauer, New York.
Furstenberg, H. (1970). Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (R. C. Gunning, ed.) 41--59. Princeton Univ. Press.
Mathematical Reviews (MathSciNet):
MR354562
Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
Mathematical Reviews (MathSciNet):
MR956646
Hajek, B. and Weller, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388--291.
Higuchi, Y. (1977). Remarks on the limiting Gibbs states on a $(d+1)$-tree. Publ. Res. Inst. Math. Sci. 13 335--348.
Mathematical Reviews (MathSciNet):
MR676482
Ioffe, D. (1996). On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137--143.
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.
Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton--Watson processes. Ann. Math. Statist. 37 1463--1481.
Mathematical Reviews (MathSciNet):
MR200979
Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337--353.
Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931--958.
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.
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.
Martinelli, F., Sinclair, A. and Weitz, D. (2004). The Ising model on trees: Boundary conditions and mixing time. Comm. Math. Phys. To appear.
Mossel, E. (2001). Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11 285--300.
Mossel, E. (2003). On the impossibility of reconstructing ancestral data and phylogenies. J. Comput. Biol. 10 669--678.
Mossel, E. (2004). Phase transitions in phylogeny. Trans. Amer. Math. Soc. 356 2379--2404.
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.
Mossel, E. and Peres, Y. (2003). Information flow on trees. Ann. Appl. Probab. 13 817--844.
Mossel, E. and Steel, M. (2004). A phase transition for a random cluster model on phylogenetic trees. Mathematical Biosciences 187 189--203.
Pemantale, R. and Peres, Y. (1995). Recursions on trees and the Ising model at critical temperatures. Unpublished manuscript.
Pemantle, R. and Steif, J. E. (1999). Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 876--912.
Semple, C. and Steel, M. (2003). Phylogenetics. Oxford Univ. Press.
Sober, E. and Steel, M. A. (2002). Testing the hypothesis of common ancestry. J. Theor. Biol. 218 395--408.
Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Probab. 3 387--398.
Mathematical Reviews (MathSciNet):
MR378152
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.
Mathematical Reviews (MathSciNet):
MR77479
