Slow mixing of local dynamics via topological obstructions
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Slow Mixing of Local Dynamics via Topological Obstructions. Dana Randall Georgia Tech. MC IND : Starting at I 0 , 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;

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Slow Mixing of Local Dynamics via Topological Obstructions

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Slow mixing of local dynamics via topological obstructions

Slow Mixing of Local Dynamics via Topological Obstructions

Dana Randall

Georgia Tech


Independent sets

MCIND:

Starting at I0,Repeat:

- Pick v Î V andb 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.

l1

l0

l2

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.

This chain connects the state space and

converges to π. How long?


Some fast mixing results

Some fast mixing results

  • Fast if  ≤ 2/(d-2) using “edge moves.”

    So for  ≤ 1 on Z2. [Luby, Vigoda]

  • Fast if  ≤ pc/(1-pc) (const for site percolation)

    i.e.,  ≤ 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 for any [Madras, Randall]

    k≤ n/2(d+1)


Sampling independent sets

Sampling: Independent Sets

Dichotomy

lsmall

l large

Sparse sets:

Fast mixing

Dense sets:

Slow mixing

Phase

Transition

l

O E O E


Slow mixing of mc ind large l

n2/2

n2/2

(n2/2-n/2)

l

l

l

S

SC

#R/#B

1

0

l large there is a “bad cut,”

. . . so MCIND is slowly mixing.x

Slow mixing of MCIND (large l)

(Even)

(Odd)


Ind sets in 2 dimensions

Ind sets in 2 dimensions

Conjecture: Slow for  > 3.79

[BCFKTVV]: Slow for  > 80 (torus)

New: Slow for  > 8.07 (grid)

> 6.19 (torus)


Slow mixing of mc ind large l1

n2/2

(n2/2-n)

n2/2

l

l

l

SC

S

Si

1

0

π(Si) = ∑|I|/Z

IÎSi

Entropy Energy

Slowmixing of MCInd: large l

#R/#B


Group by of fault lines

Def:A monochromatic bridge

is an occupied path on the

odd or even sub-lattice. A

monochromatic cross is a

bridge in both directions.

Group by # of “fault lines”

Def: Fault lines are vacant

paths of width 2 “zig-zagging”

from top to bottom (or left

to right).

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

Lying a little….

“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 lines1

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

2. Shift right

of fault by 1

and flip colors.

(FJ (I, r)Î)

(IÎFJ)

FJ : FJ x {0,1}n+l

3. Remove rt

column J; add

points along

fault line

according to r

1. Identify horizontal

or vertical fault line F.

Let F = UFJ

F,J

“Peierls Argument”

for first fault F of lengthL=n+2l and

rightmost column J.


Slow mixing of local dynamics via topological obstructions

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: 1 = ()

≥ (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) .


Slow mixing of local dynamics via topological obstructions

Lemma: J |J| ≤ c((1+1+4)/2)n .

F= UFJ .

F,J

Lemma: (FJ) ≤ |J| (1+)-(n+l ) .

(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….).


Slow mixing of local dynamics via topological obstructions

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-cnwhen 

Thm:(F) < p(n) e-cn when 

Cor: MCIND is slowly mixing for




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.

Slow mixing on the torus


Slow mixing of local dynamics via topological obstructions

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

Lemma: (F) ≤ (1+)-(n+l ) .

Thm:(F) < p(n) e-cn when 


Open probems

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, . . .


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