English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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-timein the high temperature regime ($\beta < 1$) has order $n\log n$, whereas themixing-time in the case $\beta > 1$ is exponential in $n$. Recently, Levin,Luczak and Peres proved that for any fixed $\beta < 1$ there is cutoff at time$[2(1-\beta)]^{-1} n\log n$ with a window of order $n$, whereas the mixing-timeat the critical temperature $\beta=1$ is $\Theta(n^{3/2})$. It is natural toask how the mixing-time transitions from $\Theta(n\log n)$ to $\Theta(n^{3/2})$and finally to $\exp(\Theta(n))$. That is, how does the mixing-time behave when$\beta=\beta(n)$ is allowed to tend to 1 as $n\to\infty$.  In this work, we obtain a complete characterization of the mixing-time of thedynamics 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. In the high temperature regime, $\beta = 1 -\delta$ for some $0 < \delta < 1$ so that $\delta^2 n \to\infty$ with $n$, themixing-time has order $(n/\delta)\log(\delta^2 n)$, and exhibits cutoff withconstant 1/2 and window size $n/\delta$. In the critical window, $\beta = 1\pm\delta$ where $\delta^2 n$ is O(1), there is no cutoff, and the mixing-time hasorder $n^{3/2}$. At low temperature, $\beta = 1 + \delta$ for $\delta > 0$ with$\delta^2 n \to\infty$ and $\delta=o(1)$, there is no cutoff, and the mixingtime has order $(n/\delta)\exp(({3/4}+o(1))\delta^2 n)$.

The mixing time evolution of Glauber dynamics for the mean-field Ising model