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Slow Mixing of Local Dynamics via Topological Obstructions

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

  2. 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?

  3. 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)

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

  5. 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)

  6. Ind sets in 2 dimensions Conjecture: Slow for  > 3.79 [BCFKTVV]: Slow for  > 80 (torus) New: Slow for  > 8.07 (grid) > 6.19 (torus)

  7. 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 ∞

  8. 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.

  9. 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.

  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. 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.

  12. 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) .

  13. 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….).

  14. 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 

  15. 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

  16. 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 

  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, . . .

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