1. Slow Mixing of Local Dynamics via Topological Obstructions
Dana Randall
Georgia Tech

2. Independent Sets l0 Goal: Given l, sample indep. set I with
prob π(I) = l|I|/Z,
where Z = ∑J l|J| is the partition fcn.
MCIND:
Starting at I0, Repeat:
- Pick v Î V and b Î {0,1};
- If v Î I, b=0, remove v w.p. min (1,l-1)
- If v  I, b=1, add v w.p. min (1,l)
if possible;
- O.w. do nothing. l0 l2 l1
This chain connects the state space and
converges to π How long?

3. Some fast mixing results
Fast if  ≤ 2/(d-2) using “edge moves.”
So for  ≤ 1 on Z [Luby, Vigoda]
Fast if  ≤ pc/(1-pc) (const for site percolation)
i.e.,  ≤ 1.24 on Z [Van den Berg, Steif]
Fast for “swap chain” on k (ind sets of
size = k), when k < n / 2(d+1).
[Dyer, Greenhill]
Fast for “swap chain” or M IND on
U k for any [Madras, Randall]
k≤ n/2(d+1)

4. Sampling: Independent Sets
Dichotomy
l small
l large
Sparse sets:
Fast mixing
Dense sets:
Slow mixing
Phase
Transition l
O E O E

5. Slow mixing of MCIND (large l)
(Even)
(Odd)
n2/2 l
(n2/2-n/2) 1 ∞ S SC
l large there is a “bad cut,”
. . . so MCIND is slowly mixing x 
#R/#B

6. Ind sets in 2 dimensions Conjecture: Slow for  > 3.79
[BCFKTVV]: Slow for  > 80 (torus)
New: Slow for  > (grid)
> (torus)

7. Slow mixing of MCInd: large l
(n 2/2-n) S SC 1
π(Si) = ∑ |I|/Z Si
IÎSi
#R/#B ∞
Entropy Energy

8. Group by # of “fault lines”
Def: Fault lines are vacant
paths of width 2 “zig-zagging”
from top to bottom (or left
to right).
Def: A monochromatic bridge
is an occupied path on the
odd or even sub-lattice. A
monochromatic cross is a
bridge in both directions.
Lemma: If there is no fault line, then there
is a monochromatic cross.
Lemma: If I has an odd cross and I’ has an
even cross, then P(I,I’)=0.

9. Lying a little…. Def: A fault line has only 0 or 1
“Alternation
point”
Def: A fault line has only 0 or 1
alternation points (and spans).
Lemma: If there is a spanning path,
then there is a fault line.

10. Group by # of “fault lines”
Fault lines are vacant paths
of width 2 from top to bottom
(or left to right). F  R B
. . . S SC

11. FJ : FJ x {0,1}n+l 
“Peierls Argument”
Let F = U FJ
F,J
for first fault F of length L=n+2l and
rightmost column J.
FJ : FJ x {0,1}n+l 
3. Remove rt
column J; add
points along
fault line
according to r
(FJ (I, r) Î )
1. Identify horizontal
or vertical fault line F.
(IÎ FJ)
2. Shift right
of fault by 1
and flip colors.

12. FJ : F,J x {0,1}n+l 
The Injection FJ
FJ : F,J x {0,1}n+l 
Note: FJ ( I, r) has |r|-|J| more points.
Lemma: (FJ) ≤ |J| (1+)-(n+l ) .
Pf: = ()
≥  (F,J (I,r))
I FJ r {0,1}n+l 
= r (I) |J| + |r|
=  (I) |J| r |r| 
= (I) |J| (1+(n+l )
= |J| (1+ )(n+l ) (FJ) .

13. nn+2i ,
F = U FJ .
F,J
Lemma: (FJ) ≤ |J| (1+)-(n+l ) .
Lemma: J |J| ≤ c((1+ 1+4)/2)n .
(Since Tn = Tn-1 + Tn-2 .)
Lemma: The number of fault lines
is bounded by
n2/2
nn+2i ,
i=0
where  is the self-avoiding walk
constant ( ≤ 2.679….).

14. Thm: (F ) < p(n) e-cn when 
Pf: (F ) = FJ (FJ)
≤ FJ |J| (1+)-(n+l )
≤ F (1+)-(n+l ) J |J|
≤ c inn+2i (1+)-(n+i)
. ((1+ 1+4)/2)n
 2 i (1+ 1+4) n
= p(n) ( ) ( )
1+1+
≤ p(n) e-cn when 
Cor: MCIND is slowly mixing for


15. Slow mixing on the torus
1. Identify horizontal
or vertical fault lines.
2. Shift part
between faults by
1 and flip colors.
3. Add points
along one fault
line, where
possible.

16. Lemma: (F) ≤ (1+)-(n+l ) .
Thm: (F ) < p(n) e-cn when 
Pf: (F ) = F (F)
≤ F (1+)-(n+l )
≤ in+2i (1+)-(n+i)
n+i
≤ n2 i
1+
≤ p(n) e-cn
when 1+,i.e.,  >6.183 n 2

17. Open Probems What happens between 1.2 and 6.19 on Z2 ?
Can we get improvements in higher
dimensions using topological obstructions?
(or improved bounds on phase transitions
indicating the presence of multiple Gibbs
states?)
Slow mixing for other problems:
Ising, colorings, . . .