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. |
Keywords
Arak process, Widom Rowlinson model, continuum magnetisation, phase transition, universality, rigorous results, Monte Carlo study
Math Review Classification
Primary 82B27
; Secondary 82B21
Last Updated
Length
23 pages
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