Ising Disks: Topology Preserving Glauber Dynamics

TL;DR

This paper introduces a topology-preserving dynamic model for contractible cubical sets, analyzing its connectivity and ergodic properties using Markov chains and topological constraints.

Contribution

It defines a novel topology-preserving Glauber dynamics model for cubical sets and proves its connectivity and ergodicity properties in the planar case.

Findings

The state space is connected in the planar case.

The Markov chain is irreducible and ergodic below a certain fugacity threshold.

The model relates to self-avoiding polygons and Eden model with topological constraints.

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.