Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

Abstract

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field. In this energy landscape there are $1$ stable configuration and $q-1$ metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as $β\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We also geometrically identify the union of all minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration.

Figures (14)

• Figure 2: Energy landscape in the case of $4$-state Potts model with positive external magnetic field around the unique stable state $\bold 1$ cutting the configurations with the energy bigger than $\Phi_\text{pos}(\bold m,\bold 1)$, $\bold m\in\mathcal{X}^m_\text{pos}=\{\bold 2,\bold 3,\bold 4\}$. This picture is simplified since there are not represented the cycles (valleys) that contain configurations with stability level smaller than or equal to $2$ (see Proposition \ref{['ricorrenzapos']}).
• Figure 3: Viewpoint from above of the energy landscape depicted in Figure \ref{['figurepositive']}. For every $\bold m\in\mathcal{X}^m_\text{pos}$, the cycle whose bottom is the stable state $\bold 1$ is deeper than the initial cycles $\mathcal{C}^\bold m_{\mathcal{X}^s_\text{pos}}(\Gamma_\text{pos}^m)$. These last cycles are depicted with circles whose diameter is smaller than the one related to the stable state $\bold 1$.
• Figure 4: Example of configurations on a $8\times 11$ grid graph displaying a vertical $s$-bridge (a), a horizontal $s$-bridge (b) and a $s$-cross (c). We color black the spins $s$.
• Figure 5: Stable tiles centered in any $v\in V$ for a $q$-Potts configuration on $\Lambda$ for any $m,r,s,t\in S\backslash\{1\}$ different from each other. The tiles are depicted up to a rotation of $\alpha\frac{\pi}{2}$, $\alpha\in\mathbb Z$.
• Figure 6: Possible clusters of spins different from $1$ by considering the stable tile depicted in Figure \ref{['figmattonellepos']}(f) and (g) for $\sigma$ centered in the vertex $v$. We color black the $m$-rectangle and gray the $r$-rectangles.

Theorems & Definitions (36)

• Proposition 4.1: Identification of the stable configuration
• Theorem \oldthetheorem: Identification of the metastable states
• Corollary 4.1: Maximum depth of a cycle in $\mathcal{X}\backslash\mathcal{X}^s_\text{pos}$
• Proposition 4.2: Estimate on the stability level
• Theorem \oldthetheorem: Recurrence property
• Theorem \oldthetheorem: Asymptotic behavior of $\tau_{\mathcal{X}^s_{\text{pos}}}^\bold m$ and mixing time
• Theorem \oldthetheorem
• Theorem \oldthetheorem: Minimal gates for the transition from $\bold m\in\mathcal{X}^m_\text{pos}$ to $\mathcal{X}^s_\text{pos}$
• Corollary 4.2
• Theorem \oldthetheorem: Minimal gates for the transition from $\bold m$ to $\mathcal{X}^m_\text{pos}\backslash\{\bold m\}$