Slow Mixing of Local Dynamics via Topological Obstructions

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Slow Mixing of Local Dynamics via Topological
Obstructions

• Dana Randall
• Georgia Tech

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Independent Sets
Goal Given l, sample indep. set I with
prob p(I) lI/Z, where
Z ?J lJ is the partition fcn.
This chain connects the state space and
converges to p. How long?
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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 lt n / 2(d1).
• 
  Dyer, Greenhill

• Fast for swap chain or M IND on
• U ?k for any ???????Madras,
  Randall
• k n/2(d1)

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Sampling Independent Sets
Dichotomy
l small
???l large
Sparse sets Fast mixing
Dense sets Slow mixing
Phase Transition
l
O E O E
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Slow mixing of MCIND (large l)
(Even)
(Odd)
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Ind sets in 2 dimensions
Conjecture Slow for ? gt 3.79 BCFKTVV
Slow for ? gt 80 (torus) New Slow for ? gt
8.07 (grid) gt 6.19
(torus)
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Slow mixing of MCInd large l
R/B
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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.
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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.
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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
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Peierls Argument
Let ?F U ?FJ
F,J
for first fault F of length Ln2l and
rightmost column J.
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The Injection ?FJ
???????????FJ ?F,J x 0,1nl
??
Note ?FJ ( I, r) has r-J more points.
Lemma ?(?FJ) ?J (1?)-(nl ) .
Pf 1 ?(?)
????????????? ?(?F,J (I,r)) I?
?FJ r? 0,1nl ???
???????r?? ?(I) ??J r
?????? ?(I) ??J ?r? ?r ???
???????(I) ??J
(1??(nl ) ??J (1 ?)(nl
) ?(?FJ) .
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?F U ?FJ .
F,J
Lemma ?(?FJ) ?J (1?)-(nl ) .
(Since Tn Tn-1 ?Tn-2 .)
Lemma The number of fault lines is
bounded by n2/2 ????????????????
??????n??n2i , i0
where ? is the self-avoiding walk
constant (? 2.679.).
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Thm ?(?F ) lt p(n) e-cn when ???????????
Cor MCIND is slowly mixing for
???????????
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Slow mixing on the torus
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Lemma ?(?F) (1?)-(nl ) .
Thm ?(?F ) lt p(n) e-cn when ???????????
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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, . . .