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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.00962 |
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Table of Contents:
- We derive a universal relation for the critical temperatures of the $q$-state Potts model based on the counting of domain-wall microstates. By balancing interface energy against configurational entropy, we show that the critical temperature is determined by the ratio of the coordination-dependent energy cost to the logarithm of a total multiplicity factor. This factor decomposes into a lattice-topological constant, representing a projection from an underlying orthogonal Euclidean space, and a term representing Markovian sampling in the $q$-dimensional state space. The framework recovers exact solutions for two-dimensional square, triangular, and honeycomb lattices and achieves sub-3\% accuracy for three-dimensional simple cubic, bcc, fcc, and diamond geometries. This approach unifies the Potts universality class into a single geometric classification, revealing that the phase transition is governed by the saturation of interface propagation through the lattice manifold and providing a predictive tool that characterizes the entire $q$-state family from a single topological calibration.