A Power Law of Order 1/4 for Critical Mean Field by Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres

By Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres

The Swendsen-Wang dynamics is a Markov chain customary via physicists to pattern from the Boltzmann-Gibbs distribution of the Ising version. Cooper, Dyer, Frieze and Rue proved that at the whole graph Kn the blending time of the chain is at so much O( O n) for all non-critical temperatures. during this paper the authors exhibit that the blending time is Q (1) in excessive temperatures, Q (log n) in low temperatures and Q (n 1/4) at criticality. additionally they offer an higher sure of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts version on any tree of n vertices

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1) and write x0 = X0 /n. 9). Since β : R+ → R we have that φ : [−1, ∞] → R. We begin with some preparations for the proof. 1. For c > 2, there exists an unique fixed point γ0 ∈ (1 − 2c , 1) of φ(x). 3) 1 φ(x) − γ0 < ≤ δ. 2. 4) E(X1 − γ0 n)2 ≤ δ(X0 − γ0 n)2 + Bn . 3. We have E X1 − φ(x0 )n | X0 ∈ [γ0 n, n] 2 = O(n) . 4. If X is distributed as the stationary distribution of the magnetization Swendsen-Wang chain, then E(X − γ0 n)2 = O(n). 5. Suppose√X0 , Y0 are two √ magnetization Swendsen-Wang chains such that X0 , Y0 ∈ [γ0 n − A n, γ0 n + A n] where A is a constant, we can couple X1 and Y1 such that X1 = Y1 with probability Ω(1) (which may depend on A).

17: By Cauchy-Schwartz P(X ≤ −ρh) ≥ E[X 2 1{X 2 ≥ρ2 h2 } ] ≤ EX 4 E1{X 2 ≥ρ2 h2 } ≤ bh4 P(X 2 ≥ ρ2 h2 ) . Hence, h2 ≤ EX 2 ≤ ρ2 h2 + E[X 2 1{X 2 ≥ρ2 h2 } ] ≤ ρ2 h2 + bh4 P(X 2 ≥ ρ2 h2 ) . We conclude that (1 − ρ2 )2 , b and the assertion follows by symmetry since P(X ≤ −ρh) = P(−X ≤ −ρh). 8. 18. Let Xt be a magnetization chain with X0 ∈ [b1 n3/4 , b2 n3/4 ] where b2 > b1 > 0 are two constants. Let τ1 be the first time that Xt ∈ [ b21 n3/4 , (b2 + b1 3/4 ]. Then there exists a constant C = C(b1 , b2 ) > 0 such that for all constant 2 )n δ > 0 we have P(τ1 ≤ δn1/4 ) ≤ Cδ 2 .

Direct corollary of the local central limit theorem of simple random walk. 5. To couple X1 and Y1 , we first apply the percolation step of the Swendsen-Wang dynamics in both chains independently. 7, with probability 1 − O( n1 ), the number of isolated points after percolation is bigger than 3e1c n in both chains. Conditioned on this, we assign each component a ± spin using the following procedure. First assign the spins of components independently in descending order of their size until there are 3e1c n components left.

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