Emergent structures in statistical mechanics

Equilibrium states of basic models of classical and quantum systems can often be understood in terms of spontaneously emergent structures.   Examples include planar Ising models’ fermionic degrees of freedom, spontaneous organization of Ising and Potts models into cliques,  whose statistics are given by the Fortuin - Kasteleyn random cluster models, and loop representations of a family of quantum spin chains.  The presentation of the models’ equilibrium states in such terms yields insights on its phase structure,  correlation functions,  and in the latter case conditions for dimerization, the emergence  of topological states, and the spontaneous formation of  Majorna spinor excitations at the spin chain’s edges.