Sharp mixing time asymptotics of Glauber dynamics for the Curie-Weiss-Potts model at low temperatures

TL;DR

This work analyzes Glauber dynamics for the Curie–Weiss–Potts model at low temperature and establishes a sharp, metastability-driven mixing-time formula. By projecting to the magnetization space, the authors isolate metastable valleys and derive a reduced Markov chain 𝔛_β whose mixing time governs the original dynamics on a metastable time-scale θ_N = 2πN e^{N D_β}, with D_β the depth of the wells. The main finding is that lim_{N→∞} T_δ^mix(σ_N^β) / θ_N equals the δ-mixing time of 𝔛_β, revealing that global mixing is controlled by inter-well transitions and that there is no cutoff in this regime. The paper develops a comprehensive metastability framework—recurrence inside wells, local mixing before exits, and rigorous model reduction—and extends the results to alternative Glauber dynamics (heat-bath, Metropolis), highlighting the robustness of the reduced-chain mechanism for slow mixing in mean-field spin systems.

Abstract

In this article, we derive a sharp mixing time estimate of the Glauber dynamics for the Curie-Weiss-Potts model in the low-temperature regime. In contrast to the high-temperature regime studied by Cuff et al. (J. Stat. Phys. 149: 432-477, 2012), in which the Gibbs measure is concentrated around the equiproportional distribution of spins, the Gibbs measure in the low-temperature regime is concentrated on multiple states, each with a dominant number of a single spin. Consequently, global mixing of the system requires sufficiently many transitions between these states. Since these transitions are well explained by the phenomenon of metastability, the theory of metastability plays a central role in the analysis of slow mixing. In particular, the sharp asymptotics for the mixing time is given by the mixing time of the limit Markov chain, which describes the metastable behavior of the dynamics, multiplied by the metastable transition time-scale. As a byproduct, we verify that it does not exhibit a cutoff phenomenon.

Figures (4)

• Figure 1.1: Graph of $F_{\beta}$ in the Curie--Weiss model where the horizontal axis reads the first coordinate of the elements in $\Xi=\{(x_{1},1-x_{1}):0\le x_{1}\le1\}$. If $\beta<2$ (left), then ${\bf e}$ is the only local (thus global) minimum which is non-degenerate. If $\beta=2$ (middle), the graph shape remains to be the same but in this case ${\bf e}$ is degenerate. Finally, if $\beta>2$ (right) then there exist two symmetric local minima ${\bf u}_{1}$ and ${\bf u}_{2}$, and their saddle point is the midpoint ${\bf e}$. In particular, $H_{\beta}=F_{\beta}({\bf e})$. The metastable valleys $\mathcal{E}_{N}^{1}$ and $\mathcal{E}_{N}^{2}$ (cf. \ref{['eq:ENk-def']}) are colored red.
• Figure 1.2: Energy landscape of the CWP model if $\beta>\beta_{1}$. The (cross-marked) middle point represents the equiproportional vector ${\bf e}$ and the $q$ outside (dot-marked) circle centers represent the local minima ${\bf u}_{1},\dots,{\bf u}_{q}$. Each gray-colored circle represents an energetic well $\mathcal{W}_{k}$, $k\in\llbracket0,q\rrbracket$, such that dark color indicates the deepest wells and light color indicates shallower wells. The saddle points with height $H_{\beta}$ are marked between the wells. In the case of $q\ge5$ and $\beta\in(\beta_{3},q)$, the $q$ saddle points ${\bf v}_{1},\dots,{\bf v}_{q}$ between $\mathcal{W}_{0}$ and $\widehat{\mathcal{W}}_{1}$ are marked as blue squares. The metastable valleys $\mathcal{E}_{N}^{k}$, $k\in\llbracket0,q\rrbracket$ (cf. \ref{['eq:ENk-def']} and \ref{['eq:EN0-def']}) are marked by red dashed lines for $q=3$. If $\beta\ge\beta_{3}$ and $q\ge4$, one should interpret that $\overline{\mathcal{W}}_{k}\cap\overline{\mathcal{W}}_{\ell}=\{{\bf u}_{k,\ell}\}\ne\emptyset$ for any $k\ne\ell$, even though ${\bf u}_{k,\ell}$ is illustrated only if $k,\ell$ are nearest neighbors (due to dimensional restriction).
• Figure 4.1: The cases of $\beta\in(\beta_{1},\beta_{2})$ (left) and $\beta\in(q,\infty)$ (right) for $q\ge5$. From each critical point ${\bf e}$, ${\bf v}_{k}$, ${\bf u}_{k,\ell}$ (if it exists), or $\bm{c}\in\mathcal{C}_{4}$, the teal arrow represents the descending trajectory constructed in Lemmas \ref{['l3']}, \ref{['l4']}, or \ref{['l5']}.
• Figure 4.2: The case of $q\ge5$ and $\beta\in(\beta_{3},q)$. The bigger well $\widehat{\mathcal{W}}_{1}$ contains $\mathcal{W}_{k}$ for all $k\in\llbracket1,q\rrbracket$.

Theorems & Definitions (73)

• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Definition 2.1: Description of Metastability
• Remark 2.2
• Proposition 2.3: LLM18
• Remark 2.4
• Proposition 2.5