Phase Transitions In Reconstruction

1. Phase Transitions In Reconstruction Yuval Peres, U.C. Berkeley

2. Markov Chains on trees • 2 generalizations of Markov property to infinite trees: • Markov random field. (~ two sided) • Markov chain on the tree (~ one sided) + 0 A B + +  + - + + - + - + - + + • XA and XB are cond. ind. given X (B finite). • ’[] is given by [(0)] £{e = u ! v} Me(u),(v) • Definitions extend to infinite trees. • For finite trees definitions are equivalent.

3. The reconstruction problem • Fix an infinite rooted tree T, an initial distribution  and Markov chains Me on the edges. • Let n = values of the chain at level n. • We want to know if n is “correlated” with 0 as n !1 • If T=1/2-line and Me=M for all e where M is ergodic, then the correlation between 0 and n decays exponentially in n. • For general trees, delicate balance between exponential decay of correlation and exponential growth of tree. • Remark: The problem is different from the standard setting in statistical physics, since we cannot set the boundary conditions. 0 0 n n

4. Formal Definition: Reconstruction problem T = an infinite rooted tree. Ln = { v : d(v,0) = n }. ns = projection/marginals of ’s on Ln (the measure conditioned on(0) = s) : ns[] =  { ’s[] :  | Ln = } The reconstruction problem is solvable if one of the following equivalent conditions hold: 9 i,j: limn !1 |in - jn|TV := limn !1 |in() - jn()|> 0. has a non trivial tailis non-extremal. + • Which properties of T and M determine solvability? 0 + + + - + + - + - + - + + L3

5. Statistical physics on trees The Ising model on the binary tree can be defined: Set σr, the root spin, to be +/- with probability ½. For all pairs of (parent, child) = (v, w), set σw = σv, with probability , otherwise σw = +/- with probability ½. This is exactly the CFN model. • Studied in statistical physics [Spitzer 75, Higuchi 77,Chayes-Chayes-Sethna-Thouless-86,Bleher-Ruiz-Zagrebnov 95, Evans-Kenyon-Peres-Schulman 2000, Ioffe 99, M 98, Haggstrom-M 2000, Kenyon-M-Peres 2001, Martinelli-Sinclair Weitz 2003, Martin 2003] + + + + - + + - - + - + + +

6. Recursive Reconstruction T = 3-ary regular tree with Me = M for all edges. Consider the recursive majority function. = Binary symmetric channel (BSC) = Ising model (no external field) • Let pn := P[n-fold rec-maj(n) = 0] . • Let (p) = (1-) p +  (1-p) and g(p) = 3(p)+32(p)(1-(p)) • p0 = 1 and pn+1 = g(pn) ) pn! ½ if and only if (1-2) > 2/3. • )reconstruction problem is solvable if < 1/6. • Von-Neumann (56) forreliable noisy-computation. • Later: Evans-Schulmann93, Steel94, Mossel98,00.

7. Spectral Reconstruction Let M be the Ising (BSC) model on a b-ary tree T. Is0correlated with f(n) = sign({(v) : v 2 Ln}) ? Theorem (Higuchi 77): limn P[0 = f(n)] > ½ if b(1-2)2 > 1. )reconstruction for ternary tree if < ½ - (1/3)1/2. Let M be any chain and T the b-ary tree Let be the 2nd eigenvalue of M in absolute value. Claim[Kesten-Stigum66]b ||2 > 1) reconstruction. Mossel-Peres03: b ||2 =1 is the threshold for reconstruction using the census. Janson-Mossel04: This is also the threshold for “robust” reconstruction (where level n is perturbed). Can replace b by “branching number” for general trees.

8. Non-reconstruction - Coupling down • Copying rule. For i =+,-: • P[i ! i] = = 1 – 2  • P[i ! Uniform] = 1–= 2  • Continuing down the tree, non-coupled elements form a branching process with parameter . + / - + / - = = + / - = = = = = = = = = = • If 2 · 1, branching process dies)coupling. • )for ¸ ¼ no reconstruction (this is not tight!) • The threshold for reconstruction is only known for Ising (BSC) model and is given by 22 = 1. • Seen:2 2 > 1 ) reconstruction (spectral argument)

9. Reconstruction for the CFN model • Thm: The reconstruction problem for the Ising model on the (b+1)-regular tree is solvable if and only ifb 2 > 1. • “Easy direction” [Higuchi 77]: prove that a certain reconstruction algorithm works when b 2 > 1. • Higuchi argument extends to general chains and general trees. • Will also show an argument from [M98] useful for phylogeny. • “Hard direction” [¸ 95]: Non-reconstruction? • 6 different proofs! • All involve a magic. • None extends to other markov models.

10. Ising Model on Binary Trees low interm. high bias bias no bias no bias bias 2  > 1 22 < 1 “typical” boundary “typical” boundary 2 2 > 1 2  < 1 Unique Gibbs measure The transition at 2 2 = 1 was proved by: Bleher-Ruiz-Zagrebnov95,Evans-Kenyon-Peres-Schulman2000,Ioffe99, Kenyon-Mossel-Peres-2001,Martinelli-Sinclair-Weitz2004.

11. Reconstruction for Markov models • So the threshold b 2 = 1 is important. • But [M-2000] it is not the threshold for the reconstruction problem. • Not even for 2 £ 2 markov chains, • Or symmetric markov chains on q symbols. • Moreover, there exists a markov chain M s.t.  = 0, but the reconstruction problem is solvable for some b. • Open: What is the threshold for q=3 Potts on binary tree?

12. Reconstruction for other Markov models • Leaving the Ising model … • Reconstruction for other models is more interesting. • The “natural” bound for reconstruction is b ||2 > 1, where  is the second eigen-value of M (in absolute value). • In “census reconstruction” we reconstruct from Yn = (Yn(i))i 2 A, where Yn(i) = # of times color i appears at the n’th level. • Theorem [M-Peres 2002]: The count reconstruction problem is solvable if b ||2 > 1 and unsolvable if b ||2 < 1.

13. Reconstruction for other Markov models • Theorem [M-Peres 2002]: The census reconstruction problem is solvable if b ||2 > 1 and unsolvable if b ||2 < 1. • Proof uses [Kesten-Stigum-66] theorem.

14. Reconstruction for Markov models • In “robust reconstruction”, instead of the n’th level, n, we are given n, where for each v at level n • n(v) = n(v) with probability , • n(v) = an independent color from a distribution  with probability 1 - . • Similar to “robust phase transition” Pemantle-Steif 99. • Easy: if b 2 > 1 then robust reconstruction is solvable for all > 0. • Theorem [Janson-M 2003]: If b 2 < 1, then for  > 0 small, robust reconstruction is unsolvable. • Same is true with br(T) instead of b.

15. Glauber dynamics for Ising models on trees Consider the following dynamics on +/- configurations on the tree. Start with a configuration σ. At rate 1: Pick a vertex v uniformly at random, and update σ(v) according to the conditional probability given {σ(w): w ~ v}. Converges to Ising distribution; but how fast? Note: It is trivial to sample from the distribution we are interested in. Note: The same process can be defined on any Markov Random Field.

16. Ising Model on Binary Trees low interm. high bias bias no bias no bias bias 2  > 1 22 < 1 “typical” boundary “typical” boundary 2 2 > 1 2  < 1 Unique Gibbs measure 2 = (n1 + 2 log2) 2 = O(1)

17. Temporal mixing spatial mixing Thm [BKMP]: • Let G be an ∞-graph of bounded degree; • (Gr) balls of radius r around o. • Consider a particle system (e.g. Ising; Coloring) on G s.t. Glauber dynamics on Gr satisfy τ2 = O(1). • Then, for any finite set A, if f is a function of (σv)v 2 A and g a function of (σv)|v| > r, then Cov(f,g) < exp(-Ω(r))Var1/2(f)Var1/2(g). • Open problem: spatial ) temporal? • MSW: Yes for trees. g r A f

18. Phylogeny • Here the tree is unknown. • Given a sequence of collections of random variables at the leaves (“species”). • Collections are i.i.d.! • Want to reconstruct the tree (un-rooted).

19. Phylogeny • Algorithmically “hard”.