Glauber Dynamics on Trees and Hyperbolic Graphs

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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 http//research.microsoft.com/mosse
l/
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Glauber dynamics for coloring

• G (V,E) a finite graph of n vertices, where all
  degrees D. Want to sample coloring with q gt D
  colors.
• Algorithm to sample proper coloring s of G
• Start with a proper coloring s.
• Repeat the following
• Pick a vertex v uniformly at random, and update
  the color s(v) to be uniformly chosen from the
  set q \ s(w) v w.
• Converge to uniform coloring but how fast?
  (Vigoda 2000). Does speed depend on the Geometry
  of G?

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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 s from V to -1,1 such that
• T 1/ß is the inverse temperature, and Z is
  the partition function.
• Both models have local constraint.

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Glauber dynamics for Ising models

• Algorithm to sample from the Gibbs dist.
• Start with a configuration s.
• Repeat the following
• Pick a vertex v uniformly at random, and update
  s(v) according to the conditional probability
  given s(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).

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Mixing and relaxation times

• Glauber dynamics defines a d.s. matrix with
  spectrum 1 gt ?1 gt gt. The spectral gap of the
  dynamics is 1-?1 The relaxation time is t2
  1/(1 - ?1).
• The total-variation distance between µ and ?
  is
• Let P(t,s) be the distribution of the dynamics
  started at s at time t. The mixing time of the
  dynamics is defined by
• In general

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General picture

• When ß is small (large number of colors),
• t2 T(n), and
• t1 T(n log n).
• When ß is large (small number of colors), mixing
  time may be large.
• In -L,L d, when ß lt ßc, the mixing time is
  exp(T(Ld-1 )) (physics literature)

n Ld
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Simple graphs

• The Ising model on the line graph has mixing time
  n log n for all ß.
• There exists ß(D,a) such that if G(V,E) is an
  a-expander, then for all ß gt ß(D,a), the mixing
  time of the Ising model on G is exp(T(n)).

n V
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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 ?, as the smallest integer for which there
  exists a labeling v1,,vn of V s.t. for all 1 lt
  k lt n, the number of edges connecting v1,,vk
  to vk1,,vn is at most ex(G).
• THMKMP For Ising-Glauber dynamics on G
• For Coloring-Glauber dynamics on G, when q gt ?
  1

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Application to Ising model in Zd

• Labeling order

• In Z1, gives t2 O(L2) at all ß (truth is O(L)).
• In Zd, d gt 1, gives
• t2 (exp(O(Ld-1))), which is correct (up to
  constant factor in the exp) when ß is large.
• Open problem Find properties of graphs which
  imply similar lower bounds.

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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 ß.

• Similarly, we prove polynomial mixing time for
  balls in graphs of hyperbolic tilings.

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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 gt ? 1 and Ising models for all ß.
• Folklore belief In the ordered phase (1 Gibbs
  measure) t2 poly(n) , in the unordered
  phase (multiple Gibbs measure) t2
  super-poly(n).
• For trees and hyperbolic tilings, when ß is
  large, we have 8 Gibbs measures but t2 poly(n)
  ???

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The Ising model on the binary tree

• The (Free)-Ising-Gibbs measure on the tree T
• Set sr, the root spin, to be /- with probability
  ½.
• For all pairs of (parent, child) (v, w), set sw
  sv, with probability 1 e, independently for
  all pairs (v,w).

-



-



-
-


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Relaxation time for the binary tree
mutual information H(s?) H(sr)) - H(sr,s?)
In KMP we prove that
Uniqueness phase transition plays no role for
relaxation. Extremality phase transition
linear / non-linear relaxation.
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Temporal mixing spatial mixing

• Thm KMP Let G be an 8-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 t2 O(Gr).
• Then, for any finite sets A,
• I((sv)v in A , (sv)v gt r) exp(-O(r)).
• Equivalently, if f is a function of
• (sv)v in A and g a function of (sv)v gt r,
• then Cov(f,g) lt exp(-O(r))Var(f) Var(g).
• Open problem spatial temporal?

g
r
A f
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Proof sketch

• We bound Efg when Ef Eg 0.
• Consider two dynamics on Gr
• Glauber dynamics where moves are conditioned on
  the boundary. Let Qtf(s) Ef(st), where st
  is s after t updates of this dynamics.
• Glauber dynamics where moves are independent of
  the boundary. Let Ptf(s) Ef(st), for this
  dynamics.
• Since g is independent of the configuration in
  Gr, Efg EQtfg Qtf2 g2.
• We know that Ptf2 t2-tf2.

If we find a way to replace Qt by Pt, for t gt
cr, we are done.
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Paths of disagreement

• It remains to estimate Ptf Qtf2.
• Note that Ptf Qtf, unless there exists a
  path
• v1,vk, with v1 gt 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 cr, for small c gt 0,
  
• f
  exp(-O(r)).
• (similar to v.d.Berg proof)

g
r
A f
2 2
2 2
2 2
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The ternary tree in low temperatures

• The exposure result, or a recursive argument
• prove that t2 poly(n), for all n and ß.
• To obtain lower bounds on t2, we find bottlenecks
  in the state space (easy part of
  Cheeger/conductance estimates)

for
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The ternary tree in low temperatures

• In order to obtain t2 gt n1O(1) in low
  temperatures
• A s majority of spins in level n of s are
  .
• In order to obtain t2 gt nT(ß) for freezing temp
• A s recursive maj of spins in level n of s
  are .

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The tree in med. high temperatures

• The analysis uses block dynamics we update
• sub-trees of up to h h(ß) 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 t2
• (This is well known, e.g. Martinelli. Not known
  if the same holds for the mixing time)

h
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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 s be s after an update. It suffices to
  construct a coupling s.t. Ed(s, ?) (1
  c/n) d(s, ?). This implies by a general principle
  (Chen), that t2 O(n).
• By the method of path coupling
  (Jerrum-Sinclair), suffices to show the
  contraction when s and t differ in one spin.

s
t
d(s,t)
In the top line all vertices differ in one spin
only
t
s
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The tree in med. high temperatures

• Let s and ? 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(s,?), may be bounded by
  Ed(s,?) for
• and (with no other boundary
  conditions).

v
v
v
v
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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 t2 in n.
• Linearity of t2 in n always implies extremality.