Critical configurations of the hard-core model on square grid graphs

TL;DR

The paper provides a geometric, isoperimetric-based characterization of the essential saddles governing metastable tunneling in the hard-core lattice gas on an even L x L toric grid under Glauber dynamics as β→∞. It introduces odd clusters, rhombi, and filling algorithms to describe minimal-energy transition pathways, and proves that the essential saddles are exactly the configurations in 𝒞⋆(e,o), partitioned into six families with precise geometric constraints. The main contribution is a complete saddle-structure description along with a novel perimeter/energy framework that yields an explicit energy barrier of Φ(e,o)−H(e)=L+1 and a detailed description of gate communication, which sharpens understanding of typical transition trajectories. The results advance metastability analysis for hard-core systems and provide tools potentially applicable to related lattice models with blocking effects, including extensions to other lattices and boundary conditions.

Abstract

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $β$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $β\to\infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations, also known as essential saddles, for this transition very rich and complex. We provide a comprehensive geometrical characterization of the set of critical configurations that are asymptotically visited with probability one. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and we show how not only their size but also their shape determines the characterization of the saddles.

Figures (14)

• Figure 1: Example of a hard-core configuration on the $14\times 14$ square grid with periodic boundary conditions. On the left, the occupied sites in $V_\mathrm{\mathbf{o}}$ (resp. in $V_\mathrm{\mathbf{e}}$) are highlighted in black (resp. in red). On the right, we depict the same configuration using a different visual convention, in which we highlight the odd clusters that the configuration has by drawing only the empty sites in $V_\mathrm{\mathbf{e}}$ (in white), the occupied sites in $V_\mathrm{\mathbf{o}}$ (in black), and a black line around each odd cluster representing its contour.
• Figure 2: An example of a configuration in $\mathcal{C}_{ir}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the left) and one in $\mathcal{C}_{gr}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the right).
• Figure 3: An example of a configuration in $\mathcal{C}_{cr}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the left) and one in $\mathcal{C}_{sb}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the right).
• Figure 4: An example of a configuration in $\mathcal{C}_{mb}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the left) and one in $\mathcal{C}_{ib}(\mathrm{\mathbf{e}},\mathrm{\mathbf{o}})$ (on the right).
• Figure 5: Schematic representation of the set of essential saddles, where we highlight with arrows between the set pairs that communicate at energy not higher than $-\frac{L^2}{2}+L+1$ and the initial cycles $\mathcal{C}_{\mathrm{\mathbf{e}}}$ and $\mathcal{C}_{\mathrm{\mathbf{o}}}$, see \ref{['sec:defaux']}. The vertical lines represent the partition of $\mathcal{X}$ in manifolds, see \ref{['eq:foliation']}.

Theorems & Definitions (39)

• Theorem 2.1: Essential saddles
• Lemma 3.1: Set of sites winds around the torus
• Lemma 3.2: Properties of rhombi
• Proposition 3.3: Formula for rhombus perimeter
• proof
• Proposition 3.4: Odd cluster expansion via filling algorithms
• Lemma 3.5
• proof
• Proposition 3.6: Perimeter-Minimal rhombi
• Corollary 3.7: Minimal perimeter