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

The Swendsen-Wang dynamics is a Markov chain primary through physicists to pattern from the Boltzmann-Gibbs distribution of the Ising version. Cooper, Dyer, Frieze and Rue proved that at the entire graph Kn the blending time of the chain is at such a lot O( O n) for all non-critical temperatures. during this paper the authors express that the blending time is Q (1) in excessive temperatures, Q (log n) in low temperatures and Q (n 1/4) at criticality. in addition they offer an top certain 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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A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics

The Swendsen-Wang dynamics is a Markov chain standard 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 such a lot 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.

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Let Xt be the magnetization chain and (Yt , Zt ) be the two-dimensional chain as described above. As usual P and π are the transition matrix and the stationary distribution of the Swendsen-Wang chain, respectively, and let P and π be the corresponding transition matrix and stationary distribution of (Yt , Zt ), respectively. By symmetry, configurations with same two-dimensional chain value have same distributions for any t. 4) t ||σPt − π||T V = ||(|G1 |, 0)P , π||T V . 2 it suffices to couple the chains (Yt , Zt ) and (Yt , Zt ) such that they meet with probability Ω(1) in time t = Θ(1).

By 2 + O(n). 18) 2 E X 1 − γ0 n ≤ E|C1+ | − γ0 n 2 |Cj− |2 + O(n). 5, we have E|C1+ | − φ(x0 )n ≤ O( n). 19) E|C1+ | − φ(x0 )n 2 + φ(x0 )n − γ0 n 2 2 E|C1+ | − φ(x0 )n φ(x0 )n − γ0 n φ(x0 )n − γ0 n 2 √ + O( n) φ(x0 )n − γ0 n + O(n). 20) E|C1+ | − γ0 n 2 ≤ δ12 |x0 − γ0 |2 n2 + |x0 − γ0 |O(n3/2 ) + O(n). If |x0 − γ0 | = O(n− 2 ), then |x0 − γ0 |n3/2 = O(n). If |x0 − γ0 |n 2 → ∞, we have |x0 − γ0 |O(n3/2 ) = o(|x0 − γ0 |2 n2 ). 20), we get 1 1 |Cj− |2 . 6. 21). 1. As a result, we have that E|C1+ | − γ0 n 2 ≤ (φ(x0 ) − γ0 )2 n2 + O(n) 6.

RANDOM GRAPH ESTIMATES which is positive and of order /m since expectation. We get that P(YT ≥ 0 | τ ≥ −2 27 ≥ 2 m/ and minimizes the above ) ≤ Ce− cT (T −2 m)2 m2 , for some c > 0 by a straightforward calculation, concluding our proof. 3. Proof of near-critical random graph theorems. 3. 8. We begin by proving the upper bound on E|C1 |. For any positive integer define by X the random variable X = v : |C(v)| ≥ . Observe that if |C1 | ≥ , then we must have that |X | ≥ |C1 |. Thus for any positive integer we have E|C1 | ≤ P(|C1 | < ) + EX .

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