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Ising Disks: Topology Preserving Glauber Dynamics

Yuliy Baryshnikov, Efe Onaran

Abstract

We introduce a dynamic model where the state space is the set of contractible cubical sets in the Euclidian space. The permissible state transitions, that is addition and removal of a cube to/from the set, are closest to Eden model with topological constraints, and, we show, are locally decidable. We prove that in the planar special case the state space is connected. We then define a continuous time Markov chain with a fugacity (tendency to grow) parameter. Using the correspondence between our model on the plane and the self-avoiding polygons, we prove that the Markov chain is irreducible (due to state connectivity), and is also ergodic if the fugacity is smaller than a threshold.

Ising Disks: Topology Preserving Glauber Dynamics

Abstract

We introduce a dynamic model where the state space is the set of contractible cubical sets in the Euclidian space. The permissible state transitions, that is addition and removal of a cube to/from the set, are closest to Eden model with topological constraints, and, we show, are locally decidable. We prove that in the planar special case the state space is connected. We then define a continuous time Markov chain with a fugacity (tendency to grow) parameter. Using the correspondence between our model on the plane and the self-avoiding polygons, we prove that the Markov chain is irreducible (due to state connectivity), and is also ergodic if the fugacity is smaller than a threshold.

Paper Structure

This paper contains 7 sections, 21 theorems, 93 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Cubical Set Definitions and Notation
  3. More Definitions and Fundamentals
  4. Clumps
  5. Planar Clumps
  6. Ising Ensemble on Clumps
  7. Markov model for

Key Result

Lemma 3.8

For any cubical set $\mathbf X$ and $X\in\mathcal{K}^d$, the community $|\mathcal{N}_{\mathbf X}(X)|$ cannot have an irregular point on its outer perimeter.

Figures (16)

  • Figure 1.1: A planar clump, generated by the Markov process defined in Section \ref{['Isingsec']} in the "over-critical regime". Also see Figure \ref{['snapshots']}.
  • Figure 3.2: Illustration of irregular points for $d=2$ and $d=3$
  • Figure 3.3: The cubes that compose the community of $X$ and its outer perimeter shown as thick blue line
  • Figure 4.4: Indent action for a cubical set in $\mathbb{R}^2$
  • Figure 5.5: Line clump.
  • ...and 11 more figures

Theorems & Definitions (69)

  • Definition 3.1: Regularity
  • Definition 3.2
  • Example 3.3
  • Remark 3.4
  • Definition 3.5: Community
  • Remark 3.6
  • Definition 3.7
  • Lemma 3.8
  • proof
  • Lemma 3.9
  • ...and 59 more