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

ISBN-10: 1470409100

ISBN-13: 9781470409104

The Swendsen-Wang dynamics is a Markov chain normal by way 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 so much O( O n) for all non-critical temperatures. during this paper the authors convey 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 higher 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 resources for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics

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Define the stopping time β by β = min{t : Nt ≤ m(1 − /2)} , then it is clear we can couple {Yt∧β } with {Wt∧β } such that the increments of the first are larger than of the latter process. This guarantees that every record minimum of the first process is also a record minimum of the second, and thus if we put γ w = max t : Wt is at a record minimum , then we can couple such that γ1{no record minima at times [ m/10, m]} ≤ γ w + m1{β≤ m/10} . 23 shows that the probability that there is a record minimum at some time between m/10 and m decays faster than m−2 provided that 3 m ≥ A log m for A large enough.

7) P A, Jτ −1 > n3/4 , Yτ −1 ≥ −ξn3/4 ≤ Dn−1/3 Kn1/4 = o(1). 2 Next suppose that Yτ −1 < −ξn3/4 . Then there is a t ∈ [0, Kn1/4 ] such that Yt ≤ −ξn3/4 . 1), for any starting location, we have P(X1 < −ξn3/4 ) ≤ P min{|C1+ (t)|, |C1− (t)|} + + j |Cj (t)| j≥2 − j |Cj (t)| + < −ξn3/4 . 9) P(X1 < −ξn3/4 ) = O(n−1/3 ). The union bound implies now that a1 P A, Jτ −1 > n3/4 , Yτ −1 < −ξn3/4 ≤ O(n−1/3 )Kn1/4 = o(1) , 2 8. 7) we conclude that {A, Xτ −1 , Yτ −1 ∈ [A−1 n3/4 , An3/4 ]} occurs with probability Ω(1) for some constant A.

Thus, we learn that the process e−αWt e−(1+ 2 t )α2 t−(1+ )α 2m + αt(1−50 (1+ )) , is a supermartinagle. Write f (t) = t − (1 + )α2 + α(1 − 50 (1 + )) − t2 (1 + )α . 2m We apply the optional stopping theorem on the stopping time τ = min{t ≥ Wt = 0} and get that Eef (τ ) ≤ 1 . m/ : Direct calculation gives that when we put α = 13 the function f attains its minimum √ on the interval [a m/ , m] at τ = a m/ for any a ∈ [1, 3 m/3]. Hence √ P a m/ ≤ τ ≤ m ≤ P(ef (τ ) ≥ ef (a m/ ) ) . 29) m/ ≤ τ ≤ m ≤ e−ca √ m 3 ≤ e−ca , 2 √ since a ≤ m 3 .

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

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