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Emergence of metastability on the hyperbolic lattice: Effects of boundary conditions

Vanessa Jacquier, Wioletta M. Ruszel

TL;DR

The paper analyzes metastability for the Ising model on finite hyperbolic lattices with minus boundary conditions under a small positive field $h$. Using Glauber dynamics and a potential-theoretic metastability framework, it identifies a unique metastable state, computes the energy barrier $\Gamma^{p,q}$, and establishes precise exit-time asymptotics $\tau_{\mathbf{+1}} \sim \exp(\beta\Gamma^{p,q})$, along with mixing-time and spectral-gap estimates. The analysis relies on a geometric description where plus-spin clusters map to polyamonds on the dual lattice, with a critical droplet comprising a ball of radius $r^*$ plus a boundary strip; these structures determine the barrier and transition pathways within the metastable regime $(h^*_1,h^*_2)$. Outside this region, the energy landscape changes and boundary effects drive different nucleation mechanisms, underscoring the nuanced role of hyperbolic geometry in relaxation dynamics and metastability.

Abstract

We investigate the Ising model on finite subgraphs of the hyperbolic lattice under minus boundary conditions and in the presence of a positive external field $h$. Interpreting the boundary as frozen or cold wall conditions, we show that, for small values of $h$, the system exhibits metastable behaviour. Our result is very surprising, since non-amenable graphs, such as hyperbolic lattices, feature exponentially growing boundaries, which typically destabilize local energy minima. In particular, we identify the unique metastable state and characterize the exit time from it. Furthermore, we establish asymptotic results for the distribution of the first hitting time and provide estimates for the spectral gap. Finally, we analyze the energy landscape and describe the nucleation mechanism for values of $h$ outside the metastable regime.

Emergence of metastability on the hyperbolic lattice: Effects of boundary conditions

TL;DR

The paper analyzes metastability for the Ising model on finite hyperbolic lattices with minus boundary conditions under a small positive field . Using Glauber dynamics and a potential-theoretic metastability framework, it identifies a unique metastable state, computes the energy barrier , and establishes precise exit-time asymptotics , along with mixing-time and spectral-gap estimates. The analysis relies on a geometric description where plus-spin clusters map to polyamonds on the dual lattice, with a critical droplet comprising a ball of radius plus a boundary strip; these structures determine the barrier and transition pathways within the metastable regime . Outside this region, the energy landscape changes and boundary effects drive different nucleation mechanisms, underscoring the nuanced role of hyperbolic geometry in relaxation dynamics and metastability.

Abstract

We investigate the Ising model on finite subgraphs of the hyperbolic lattice under minus boundary conditions and in the presence of a positive external field . Interpreting the boundary as frozen or cold wall conditions, we show that, for small values of , the system exhibits metastable behaviour. Our result is very surprising, since non-amenable graphs, such as hyperbolic lattices, feature exponentially growing boundaries, which typically destabilize local energy minima. In particular, we identify the unique metastable state and characterize the exit time from it. Furthermore, we establish asymptotic results for the distribution of the first hitting time and provide estimates for the spectral gap. Finally, we analyze the energy landscape and describe the nucleation mechanism for values of outside the metastable regime.

Paper Structure

This paper contains 20 sections, 11 theorems, 88 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Model description and analysis of the energy landscape
  3. Ising model on $\mathcal{L}_{p,q}$
  4. Glauber dynamics
  5. Metastability problem
  6. Metastable region
  7. Main results
  8. Geometry of the model: clusters and polyamonds
  9. Proofs of main results
  10. Proof of Proposition \ref{['teoRP']}
  11. Analysis of the configurations in $R_1$.
  12. Analysis of the configurations in $R_2$.
  13. Proof of Theorem \ref{['Identification']}
  14. Proof Theorems \ref{['thm:transitiontime']}, \ref{["teotime'"]} and \ref{['TMIX']}
  15. Discussion on the other regions
  16. ...and 5 more sections

Key Result

Theorem 1

(Identification of metastable states) For $h$ in some specific critical region, the metastable state is equal to the homogeneous state $\textbf{-1}$ and the maximal stability level $\Gamma^{p,q}$ is an explicit constant depending on $p,q$.

Figures (16)

  • Figure 1: Schematic depiction of the energy landscape in the critical region.
  • Figure 2: Embedding of $\mathcal{L}_{4,5}$ in the hyperbolic disc
  • Figure 3: $\mathcal{L}_{5,4}$ as the dual lattice of $\mathcal{L}_{4,5}$.
  • Figure 4: Example of a critical droplet with radius $r^*=1$ and strip $S_{crit}(h)$ in the layer $L_1$ of length $k(h)=2$.
  • Figure 5: Example of a configuration on $B_{5;4,5}(\textbf{o})$. Blue vertices represent plus spins and the remaining non-colored vertices $-1$ spins.
  • ...and 11 more figures

Theorems & Definitions (30)

  • Theorem
  • Proposition
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lem:reversibility']}
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 20 more