1. Glauber Dynamics on Trees and Hyperbolic Graphs
Elchanan Mossel, Microsoft Research
joint work with
Claire Kenyon, L.R.I. Paris IX
Yuval Peres, U.C. Berkeley
11/3/2018

2. Glauber dynamics for coloring
G = (V,E) a finite graph of n vertices, where all degrees ≤ D. Want to sample coloring with q > D colors.
Algorithm to sample “proper” coloring σ of G
Start with a proper coloring σ.
Repeat the following:
Pick a vertex v uniformly at random, and update the color σ(v) to be uniformly chosen from the set [q] \ {σ(w): v ~ w}.
Converge to uniform coloring; but how fast? (Vigoda 2000). Does speed depend on the Geometry of G?
11/3/2018

3. The Ising Model It is easier to analyze the Ising model:
Let G=(V,E) be a finite graph with n vertices.
The Ising model on G is a probability measure (Gibbs distribution) on the space of configurations σ from V to {-1,1} such that
T = 1/β is the “inverse temperature”, and Z is the “partition function”.
Both models have “local” constraint.
11/3/2018

4. Glauber dynamics for Ising models
Algorithm to sample from the Gibbs dist.
Start with a configuration σ.
Repeat the following:
Pick a vertex v uniformly at random, and update σ(v) according to the conditional probability given {σ(w): w ~ v}.
Converge to Gibbs distribution; but how fast? Does speed depend on the Geometry of G?
Mostly studied when G is a box in Zd (Martinelli lecture notes).
11/3/2018

5. Mixing and relaxation times
Glauber dynamics defines a d.s. matrix with spectrum 1 > γ1 > … >. The “spectral gap” of the dynamics is 1-γ1 The “relaxation time” is τ2 = 1/(1 - γ1).
The “total-variation” distance between μ and ν is:
Let P(t,σ) be the distribution of the dynamics started at σ at time t. The “mixing time” of the dynamics is defined by:
In general:
11/3/2018

6. General picture When β is small (large number of colors),
τ2 = Θ(n), and
τ1 = Θ(n log n).
When β is large (small number of colors), mixing time may be large.
In [-L,L] d, when β < βc, the mixing time is exp(Θ(Ld-1 )) (physics literature)
n = Ld
11/3/2018

7. Simple graphs
The Ising model on the line graph has mixing time n log n for all β.
There exists β(D,α) such that if G=(V,E) is an α-expander, then for all β > β(D,α), the mixing time of the Ising model on G is exp(Θ(n)).
n = |V|
11/3/2018

8. Bounding relaxation by exposure
Following the “canonical path” method of Jerrum-Sinclair (& Martinelli), We define the exposure, ex(G) of a graph G=(V,E) of maximal degree Δ, as the smallest integer for which there exists a labeling v1,…,vn of V s.t. for all 1 < k < n, the number of edges connecting {v1,…,vk} to {vk+1,…,vn} is at most ex(G).
THM[KMP]: For Ising-Glauber dynamics on G:
For Coloring-Glauber dynamics on G, when q > Δ + 1:
11/3/2018

9. Application to Ising model in Zd
Labeling order
In Z1, gives τ2 = O(L2) at all β (truth is O(L)).
In Zd, d > 1, gives
τ2 = (exp(O(Ld-1))), which is correct (up to constant factor in the exp) when β is large.
Open problem: Find properties of graphs which imply similar lower bounds.
11/3/2018

10. Trees and hyperbolic graphs
For the binary tree T, using DFS order, ex(T) is the height of T, and therefore the relaxation (mixing) time is poly(|T|) for all β.
Similarly, we prove polynomial mixing time for balls in graphs of hyperbolic tilings.
11/3/2018

11. Remarks
For trees, it easy to generate the Gibbs distribution rapidly, in a top-bottom manner.
For hyperbolic graphs, our results give a polynomial time algorithm, for sampling colorings when q > Δ + 1; and Ising models for all β.
Folklore belief: In the “ordered phase” (1 Gibbs measure) – τ2 = poly(n) , in the “unordered phase” (multiple Gibbs measure) – τ2 = super-poly(n).
For trees and hyperbolic tilings, when β is large, we have ∞ Gibbs measures but τ2 = poly(n) ???
11/3/2018

12. The Ising model on the binary tree
The (Free)-Ising-Gibbs measure on the tree T:
Set σr, the root spin, to be +/- with probability ½.
For all pairs of (parent, child) = (v, w), set σw = σv, with probability 1 – ε, independently for all pairs (v,w). + + + + - + + - + - + - + + +
11/3/2018

13. Relaxation time for the binary tree
In KMP we prove that:
mutual information:
H(σ∂) + H(σr)) - H(σr,σ∂)
Temp
1 - 2ε
σr | σ∂≡ 1
Uniq
I(σr,σ∂)
Free measure τ2
high
< 1/2
unbiased V
→ 0
extremal
O(n)
med.
(1/2,1/√2)
biased X
low
> 1/√2
Inf > 0
Non-ext
n1+Ω(1)
freeze
1 – o(1)
nΘ(β)
Uniqueness phase transition plays no role for relaxation.
Extremality phase transition linear / non-linear relaxation.
11/3/2018

14. Temporal mixing spatial mixing
Thm [KMP]: Let G be an ∞-graph of bounded degree; (Gr) balls of radius r around o. Consider nearest-neighbor particle system (e.g. Ising; Coloring) on G s.t. Glauber dynamics on Gr satisfy τ2 = O(|Gr|).
Then, for any finite sets A,
I((σv)v in A , (σv)|v| > r) = exp(-Ω(r)).
Equivalently, if f is a function of
(σv)v in A and g a function of (σv)|v| > r,
then Cov(f,g) < exp(-Ω(r))Var(f) Var(g).
Open problem: spatial temporal? g r A f
11/3/2018

15. If we find a way to replace Qt by Pt, for t > c*r, we are done.
Proof sketch
We bound E[f*g] when E[f] = E[g] = 0.
Consider two dynamics on Gr:
Glauber dynamics where moves are conditioned on the boundary. Let Qt[f](σ) = E[f(σt)], where σt is σ after t updates of this dynamics.
Glauber dynamics where moves are independent of the boundary. Let Pt[f](σ) = E[f(σt)], for this dynamics.
Since g is independent of the configuration in Gr, E[f*g] = E[Qt[f]*g] ≤ |Qt[f]|2 |g|2.
We know that: |Pt[f]|2 ≤ τ2-t|f|2.
If we find a way to replace Qt by Pt, for t > c*r, we are done.
11/3/2018

16. Paths of disagreement It remains to estimate |Ptf – Qtf|2.
Note that Pt[f] = Qt[f], unless there exists a path
v1,…vk, with |v1| > r and vk in A, s.t. vi is updated after vi-1,and vk is updated before time t.
Since all updates are
contractions in L2:
|Ptf – Qtf| ≤ |f| P[ ]
When t = c*r, for small c > 0,
= |f| exp(-Ω(r)).
(similar to v.d.Berg proof) g r A f 2 2 2
11/3/2018

17. The ternary tree in low temperatures
The exposure result, or a recursive argument
prove that τ2 = poly(n), for all n and β.
To obtain lower bounds on τ2, we find bottlenecks in the state space (easy part of Cheeger/conductance estimates):
for
11/3/2018

18. The ternary tree in low temperatures
In order to obtain τ2 > n1+Ω(1) in low temperatures
A = {σ : majority of spins in level n of σ are +}.
In order to obtain τ2 > nΘ(β) for freezing temp:
A = {σ : recursive maj of spins in level n of σ are +}.
11/3/2018

19. The tree in med. & high temperatures
The analysis uses “block dynamics”: we update
sub-trees of up to h = h(β) levels, which
include all sub-trees of h levels,
and all sub-trees of ≤ h levels which
contain leaves, or the root.
The block dynamics and the single-site dynamics have up to a constant (which depends on h) the same τ2
(This is well known, e.g. Martinelli. Not known if the same holds for the mixing time) h
11/3/2018

20. The tree in med. & high temperatures
We define a weighted hamming metric for the b-ary tree: , where |v| = distance from v to the root.
Let σ’ be σ after an update. It suffices to construct a coupling s.t. E[d(σ’, η’)] ≤ (1 – c/n) d(σ, η). This implies by a general principle (Chen), that τ2 = Ω(n).
By the method of “path coupling” (Jerrum-Sinclair), suffices to show the contraction when σ and τ differ in one spin. σ
d(σ,τ) τ
In the top line all vertices differ in one spin only σ’ τ’
11/3/2018

21. The tree in med. & high temperatures
Let σ and η differ at a single site v. There are 4 cases to consider, depending on the relative location of the updated block and v:
It turns out that for the Ising model, in the last two cases Ed(σ’,η’), may be bounded by Ed(σ’,η’) for
and (with no other boundary conditions). v v v v
11/3/2018

22. Summary
We show how the exposure of a graph gives an upper bound on the relaxation time for Glauber dynamics for Ising models and colorings of the graph.
For trees and hyperbolic graphs, the relaxation time is always polynomial in the size of the graph.
For the tree, the uniqueness phase-transition plays no role for the relaxation time, and the extremality phase transition corresponds to linearity of τ2 in n.
Linearity of τ2 in n always implies extremality.
11/3/2018