Spontaneous magnetisation in the plane

TL;DR

This paper introduces a new planar model based on the Arak process, demonstrating a phase transition at non-zero temperature and showing that its critical behavior aligns with the two-dimensional Ising model.

Contribution

It establishes the Gibbs distribution for the Arak process, introduces a temperature-parameterized model, and proves the existence of a phase transition with Ising-like critical exponents.

Findings

Confirmed phase transition at non-zero temperature.

Critical exponents match those of the 2D Ising model.

Simulation results support theoretical predictions.

Abstract

The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions form a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.