By Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres
The Swendsen-Wang dynamics is a Markov chain universal by means of 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 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 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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Additional info for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics
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Let Xt and Yt be two magnetization chains such that Xt starts from an arbitrary location and Yt starts from the stationary distribution. To prove an upper bound of order n1/4 to the mixing time we show that we can couple Xt and Yt so that they meet in time O(n1/4 ) with probability Ω(1). 3. Consider the following slight modification to the magnetization chain Xt . Instead of choosing a random spin for each component after the percolation step, we assign a positive spin to the largest component and random spins for all other components.
We get that Eτ 2 ≥ 2mEY m − o(1) . 14). 26 we deduce the same estimate for EY m . This yields that Eτ 2 ≥ 2 m2 − C 3 m2 , for some C > 0. 37) gives that for some C > 0 we have Var(τ ) ≤ E (τ − m)2 ≤ Cm . We conclude Eτ = Eτ 2 − Var(τ ) ≥ m C 1−C − 3m ≥ m − C 2m − C −2 , √ since 1 − x ≥ 1 − x for x ∈ (0, 1). 35) finishes the proof. 9. 38) valid for any A satisfying 1 ≤ A ≤ stating that √ 3 m. 25 P(|C(v)| ≥ 2 m + A m/ ) = O( e−cA ) . 2 Write X = |{v : |C(v)| ≥ 2 m + A m/ }| so that EX = O( me−cA ).
16 with T = = 20 m and get by the previous display that 3 8 P(|C(v)| ≥ m/20) ≤ 2 − 2 + O( 3 ) + C1 −7/2 m−3/2 e− m/4 3 8 = 2 − 2 + O( 3 ) , 3 as long as 3 m ≥ A log m for large enough A. 34) concludes the proof of the upper bound on E|C1 |. We turn to the proof of the lower bound on E|C1 |. Recall that at each record minimum of the process {Yt } we are starting the exploration of a new component. 35) |C1 | ≥ m − γ + τ . < 0} . 28 YUN LONG, ASAF NACHMIAS, WEIYANG NING, and YUVAL PERES Thus, in order to complete the proof we will provide an upper bound on Eγ and a lower bound on Eτ .