Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field

Abstract

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.

Figures (15)

• Figure 1: Example of configurations belonging to the set of the minimal-restricted gates between the stable configurations $\mathbf r$ and $\mathbf s$. We color white the vertices with spin $r$, gray those vertices with spin $s$.
• Figure 2: Examples of configurations which belong to $\bar{R}_{3,8}(r, s)$ (a), $\bar{B}_{4,7}^4(r, s)$ (b) and $\tilde{\mathcal{B}}_{6,9}^{6}(r, s)$ (c). For semplicity we color white the vertices whose spin is $r$ and we color gray the vertices whose spin is $s$.
• Figure 3: Example of configuations belonging to $\overline{\mathscr P}(\mathbf r,\mathbf s)$ and $\overline{\mathcal{Q}}(\mathbf r,\mathbf s)$ on a $9\times 12$ grid $\Lambda$. Gray vertices have spin value $s$, white vertices have spin value $r$. By flipping to $r$ a spin $s$ among those with the lines, the path enters into $\bar{B}_{1,K}^{K-2}(r,s)\subset\overline{\mathcal{Q}}(\mathbf r,\mathbf s)$; instead, by flipping to $r$ a spin $s$ among those with dots, the path goes to $\bar{R}_{2,K-1}(r,s)\subset\overline{\mathcal{Q}}(\mathbf r,\mathbf s)$.
• Figure 4: Example of configuations belonging to $\overline{\mathscr H}(\mathbf r,\mathbf s)$ on a $9\times 12$ grid $\Lambda$. White vertices have spin $r$, gray vertices have spin $s$. By flipping a spin $s$ to $r$ both among those with dots and among those with lines, the path can pass to another configuration belonging to $\overline{\mathscr H}(\mathbf r,\mathbf s)$.
• Figure 5: Focus on the energy landscape between $\mathbf r$ and $\mathbf s$ and example of some essential saddles for the transition $\mathbf r\to\mathbf s$ following an optimal path which does not pass through other stable states.

Theorems & Definitions (29)

• Proposition 2.1: Identification of the stable configurations
• Theorem \oldthetheorem: Minimal-restricted gates
• Theorem \oldthetheorem: Union of all minimal-restricted gates
• Corollary 3.1: Crossing the gates
• Theorem \oldthetheorem: Minimal gates for the transition $\mathbf r\to\mathcal{X}^s\backslash\{\mathbf r\}$
• Theorem \oldthetheorem: Union of all minimal gates for the transition $\mathbf r\to\mathcal{X}^s\backslash\{\mathbf r\}$
• Remark 3.1
• Corollary 3.2: Crossing the gate
• Theorem \oldthetheorem: Minimal gates for the transition $\mathbf r\to\mathbf s$
• Theorem \oldthetheorem: Union of all minimal gates for the transition $\mathbf r\to\mathbf s$