Slow Mixing of Local
Dynamics via Topological
Obstructions
Dana Randall
Georgia Tech
Independent Sets
Goal: Given l, sample indep. set I with
prob
π(I) = l|I|/Z,
where Z = ∑J l|J| is the partition fcn.
MCIND:
l0
l1
l2
Starting at I0, Repeat:
- Pick v  V and b  {0,1};
- If v  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.
This chain connects the state space and
converges to π.
How long?
Some fast mixing results
 Fast if l ≤ 2/(d-2) using “edge moves.”
So for l ≤ 1 on Z2.
[Luby, Vigoda]
 Fast if l ≤ pc/(1-pc) (const for site percolation)
i.e., l ≤ 1.24 on Z2
[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
k≤ n/2(d+1)
for any l
[Madras, Randall]
Sampling: Independent Sets
Dichotomy
l small
l large
Sparse sets:
Fast mixing
Dense sets:
Slow mixing
Phase
Transition
O
E
O
E
l
Slow mixing of MCIND
(large l)
(Even)
n2/2
l
S
(Odd)
(n2/2-n/2)
n2/2
l
l
SC
1
0
l large
there is a “bad cut,”
. . . so MCIND is slowly mixing.
∞
#R/#B
x
Ind sets in 2 dimensions
Conjecture: Slow for l > 3.79
[BCFKTVV]: Slow for l > 80 (torus)
New: Slow for l > 8.07 (grid)
> 6.19 (torus)
Slow mixing of MCInd: large l
n 2/2
(n 2/2-n)
l
l
l
SC
S
0
n 2/2
Si
π(Si) =
∞
1
∑
ISi
l|I|/Z
Entropy Energy
#R/#B
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.
Lying a little….
QuickT
im e™an
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TIFF U
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r e.
“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.
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
“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
(FJ (I, r)  )
(I FJ)
1. Identify
horizontal
or vertical
fault line F.

2. Shift right
of fault by 1
and flip colors.
3. Remove rt
column J; add
points along
fault line
according to r
The Injection FJ
FJ : F,J
Note:
x {0,1}n+l
FJ ( I, r)
has |r|-|J| more points.
Lemma:
(FJ)
Pf:
()
1 =
≥

≤ l|J| (1+l)-(n+l ) .
 (F,J (I,r))
I FJ r {0,1}n+l
=
r (I) l-|J| + |r|
=

=
(I)
=
l-|J| (1+ l)(n+l )
(I)
l-|J|
r
l|r|
l-|J| (1+l(n+l )
(FJ) .
F = U FJ .
F,J
Lemma:
(FJ)
Lemma:
J l|J| ≤
≤ l|J| (1+l)-(n+l ) .
c((1+ 1+4l)/2)n .
(Since Tn = Tn-1 + lTn-2 .)
Lemma: The number of fault lines
is bounded by
n2/2

i=0
nn+2i ,
where  is the self-avoiding walk
constant ( ≤ 2.679….).
Thm:
Pf:
(F ) < p(n) e-cn
when l
(F ) = FJ (FJ)
≤
FJ
≤
F (1+l)-(n+l ) J l|J|
≤ c
l|J| (1+l)-(n+l )
i nn+2i (1+l)-(n+i)
. ((1+
2
i
1+4l)/2)n
(1+ 1+4l)
( )(
= p(n) 1+l21+l
≤ p(n) e-cn
Cor:
n
)
when l
MCIND is slowly mixing for
l
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.
Lemma:
Thm:
Pf:
(F)
≤ (1+l)-(n+l ) .
(F ) < p(n) e-cn
when l
(F ) = F (F)
≤
F (1+l)-(n+l )
≤
n
2
i2n+2i (1+l)-(n+i)
≤ n2
2n+i
i 
1+l
≤ p(n) e-cn
when 1+l ,
i.e., l > 6.183
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, . . .