We study the Curie-Weiss model, i.e., Glauber dynamics for the Ising model on the complete graph on $n$ vertices. It is well-known that the mixing-time in the high temperature regime has order $n\log n$ [Aizenman and Holley (1984)], whereas the mixing-time in low temperatures is exponential in $n$ [Griffiths et al. (1966)]. Recently, Levin, Luczak and Peres proved that for any fixed high temperature there is cutoff, whereas the mixing-time at the critical temperature $\beta=1$ has order $n^{3/2}$. It is natural to ask how the mixing-time transitions from order $n\log n$ (with cutoff) to order $n^{3/2}$ and finally to $\exp\left(\Theta(n)\right)$. That is, how does the mixing-time behave when the inverse-temperature $\beta=\beta(n)$ is allowed to tend to $1$ as $n$ tends to infinity.

In this work, we extend the results of Levin et al. into a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point $\beta_c=1$. In particular, we find a scaling window of order $1/\sqrt{n}$ around the critical temperature. Cutoff occurs only in the subcritical regime, where we determine the cutoff point and window. Furthermore, for each temperature we also determine the order of the spectral gap of the Glauber dynamics.

A key element in the proofs is the analysis of the magnetization chain (the sum of all spins), as it turns out that the mixing of this birth-and-death chain essentially dictates the mixing of entire dynamics. If time permits, I will also present a related general result on the criterion for total-variation cutoff in birth-and-death chains (analogous to the result on convergence in separation by [Diaconis and Saloff-Coste (2006)]).