Spontaneous magnetisation in the plane

Geoff Nicholls


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.

Arak process, Widom Rowlinson model, continuum magnetisation, phase transition, universality, rigorous results, Monte Carlo study

Math Review Classification
Primary 82B27 ; Secondary 82B21

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23 pages

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