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

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?

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 Dichotomy l small l large Sparse sets: Fast mixing Dense sets: Slow mixing Phase Transition l O E O E

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

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

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…. 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.

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

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.

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: 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) .

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….).

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 

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: (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

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, . . .