The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

• Published in: Communications in mathematical physics Vol. 289; no. 2; pp. 725 - 764
• Main Authors: Ding, Jian, Lubetzky, Eyal, Peres, Yuval
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2009
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; High Temperature Regime; Stationary Distribution; Ising Model; Markov Chain; Transition Kernel
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-009-0781-9

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ( β < 1) has order n log n , whereas the mixing-time in the case β > 1 is exponential in n . Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n , whereas the mixing-time at the critical temperature β = 1 is Θ( n 3/2 ). It is natural to ask how the mixing-time transitions from Θ( n log n ) to Θ( n 3/2 ) and finally to exp (Θ( n )). That is, how does the mixing-time behave when β =  β ( n ) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 <  δ < 1 so that δ 2 n → ∞ with n , the mixing-time has order ( n / δ ) log( δ 2 n ), and exhibits cutoff with constant and window size n / δ . In the critical window, β = 1± δ , where δ 2 n is O (1), there is no cutoff, and the mixing-time has order n 3/2 . At low temperature, β = 1 +  δ for δ > 0 with δ 2 n → ∞ and δ =  o (1), there is no cutoff, and the mixing time has order .