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 $${\frac{1}{2(1-\beta)}n\log n}$$ 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 $${1/\sqrt{n}}$$ 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 $${\frac{1}{2}}$$ 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 $${\frac{n}{\delta}{\rm exp}\left((\frac{3}{4}+o(1))\delta^2n\right)}$$ .

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c , the inverse-gap is O(1) for β < β c , polynomial in the surface area for β = β c and exponential in it for β > β c . This has been proved for $${\mathbb{Z}^2}$$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β c and exponential for β > β c were established, where β c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β c , the inverse-gap and mixing-time are both exp[Θ((β − β c )h)].

We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature β c =1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ 2 n→∞, then the mixing-time has order (n/δ)log (δ 2 n). The cutoff constant is (1/2+[2(ζ2 β/δ−1)]−1), where ζ is the unique positive root of g(x)=tanh (β x)−x, and the cutoff window has order n/δ.