Spontaneous magnetisation in the plane
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 nonintersecting 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 nonzero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this offlattice model and find they are consistent with those of the Ising model in two dimensions.
 Publication:

arXiv eprints
 Pub Date:
 July 2000
 arXiv:
 arXiv:condmat/0007303
 Bibcode:
 2000cond.mat..7303N
 Keywords:

 Statistical Mechanics;
 Probability
 EPrint:
 23 pages numbered 1,0...21, 8 figures