Topological Phase Transition in Plaquette Random Cluster Model — E8 Intelligence Research

FINDING: The Plaquette Random Cluster Model bridges Potts lattice gauge theory and random cell-complex homology, revealing a topological phase transition governed by the geometry of plaquette cycles rather than local spin correlations. | MATH: The model defines a probability measure on plaquette subsets of a cell complex: \\(P(A) \\propto q^{k(A)} v^{|A|}\\), where \\(A\\) is a set of plaquettes, \\(k(A)\\) is the number of connected components of the dual complex, \\(q\\) is the number of Potts colors (gauge group order), and \\(v\\) is the plaquette fugacity. The homology enters via the boundary operator \\(\\partial_2: C_2 \\to C_1\\); the critical surface satisfies \\(v_c(q) = \\frac{1}{\\sqrt{q}} + O(1/q)\\) in the large-\\(q\\) limit, mirroring the random-cluster model's \\(p_c = \\frac{1}{1+\\sqrt{q}}\\). The topological order parameter is the rank of \\(H_1\\) (first homology) of the plaquette complex, which jumps discontinuously at the transition. | CONNECTION: The critical fugacity \\(v_c = 1/\\sqrt{q}\\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com