1. Phase Transitions In Reconstruction Yuval Peres, U.C. Berkeley
11/13/2018

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) + 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.
11/13/2018

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
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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? + + + - + +
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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] + + + + - + +
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6. Recursive Reconstruction
= Binary symmetric channel (BSC)
= Ising model (no external field)
T = 3-ary regular tree with Me = M for all edges.
Consider the recursive majority function.
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) for reliable noisy-computation.
Later: Evans-Schulmann93, Steel94, Mossel98,00.
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7. Spectral Reconstruction
Let M be the Ising (BSC) model on a b-ary tree T.
Is 0 correlated 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.
11/13/2018

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)
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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 if b 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.
11/13/2018

10. Ising Model on Binary Trees
low
interm.
high
bias
bias
no bias
bias
no bias
“typical”
boundary
2 2 > 1
“typical”
boundary
2  > 1
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.
11/13/2018

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?
11/13/2018

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.
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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.
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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.
11/13/2018

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.
11/13/2018

16. Ising Model on Binary Trees
low
interm.
high
bias
bias
no bias
bias
no bias
“typical”
boundary
2 2 > 1
“typical”
boundary
2  > 1
22 < 1
2  < 1
Unique Gibbs measure
2 = (n1 + 2 log2  )
2 = O(1)
11/13/2018

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
11/13/2018

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).
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19. Phylogeny
Algorithmically “hard”.
11/13/2018