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

The Swendsen-Wang dynamics is a Markov chain universal by way of 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 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 supply 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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**Extra resources for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics**

**Example text**

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.

2 2 5. 22 shows that P(T ≤ 2 m) ≤ m−2 , and so EXτ ∧T = o(1) and E(τ ∧T )2 = Eτ 2 + o(1). 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 ) (1+ ) + 55 2 (t ∧ T )} is a submartingale, where T Thus the process {Xt∧T + (t∧T 2m is deﬁned as T = min{t − m : t ≥ m, Nt ≤ m − t − 50 2 m} . Optional stopping yields that 2m 110 2 m (EX0 − EXτ ∧T ) − E[τ ∧ T ] . 36) E(τ ∧ T )2 ≥ P(τ < m − a m/ ) ≤ e−ca . 23, one can deduce P(τ > m + a m/ ) ≤ P(τ > m + a m/ , Yt > 0 for t ∈ [ 2 a m/ ) + e−ca 2 2 > 2 m) + e−ca , /2 a 2 m/ , m]) + e−ca ≤ P(|C1 | > 2 m + ≤ P(X2 ma √ m/ where X2 ma√m/ /2 is the number of vertices v such that |Cv | ≥ 2 m + deﬁned in the beginning of the proof.