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

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

The Swendsen-Wang dynamics is a Markov chain commonly used 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 so much 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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Extra info for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics

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5 m) and bound the rest by E( j≥1 |Cj |2 )2 (instead of j ≥ 2). 12 shows that E( j≥1 |Cj |2 )2 is controlled by (E j≥1 |Cj |2 )2 . 5 E( m} ≤ Ce−c 3 m m4 4 = O(m2 −2 ). 12 to estimate the remaining subcritical graph. This is done identically to part (ii) and we omit the details. 14. We will use a second moment method. First we show that |Cj |2 ≥ cm5/4 , E √ |Cj |≤δ m for some c = c(δ) > 0. 41) √ δ√ δ√ mP( m ≤ |C(v)| ≤ δ m). 2 2 We proceed further by restricting to the case that Cv is tree. Indeed, we have E|C(v)|1{|C(v)|≤δ√m} ≥ E|C(v)|1{ δ √m≤|C(v)|≤δ√m} ≥ 2 √ δ√ P( m ≤ |C(v)| ≤ δ m) ≥ 2 = √ δ m √ k=δ/2 m √ δ m P(|C(v)| = k, C(v) is a tree) √ k=δ/2 m k m − 1 k−2 k−1 p (1 − p)k(m−k)+(2)−(k−1) .

10 we have E|Cj+ |2 ≤ C2 n x0 n −1 E|Cj+ |4 ≤ C4 n x0 n −5 E|Cj− |2 ≤ C2 n x0 n −1 E|Cj− |4 ≤ C4 n x0 n −5 j≥2 and j≥2 = O(n5/4 ) = O(n9/4 ). 12, we have j≥1 and j≥1 = O(n5/4 ) = O(n9/4 ). 13 to handle the cross terms finishes the proof of part (ii) of the lemma. 12, n − x0 n n |Cj− |2 ≥ c2 ≥ c2 A−1 n1/4 ≥ cn5/4 . 17. Let X be a real valued random variable with EX = 0 and EX 2 ≥ h and EX 4 ≤ bh4 where b ≥ 1. Then for any ρ ∈ [0, 1] we have 2 (1 − ρ2 )2 . 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 ) .

This concludes the proof since any lower bound of the mixing time of Xt implies the same lower bound of mixing time of σt . 6. 11) for all x > 2 c 2φ(x) x+1 − 1. 12) 2(x + 1)φ − 2φ . (x + 1)2 1 − 2e−cφ + e−2cφ . 12), we have that 12 < φ if and only if e−cφ > 1 − cφ which is true for all cφ > 0. 13) which holds for all cφ > 0. Since c > 2 (which implies 2c − 1 < 0), we have that φ < 1 for all x ∈ [0, 1]. Since φ is continuous, we have a constant δ1 ∈ (0, 1) such that φ (x) < δ1 for all x ∈ [0, 1].

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