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Complex systems are networks in which very simple interactions at the microscale level determine the macroscale properties of the system, and where the macroscale behavior is not necessarily continuously dependent on the parameters of the system. The Potts model is a rapidly emerging and increasingly applicable predictive statistical mechanics model for complex systems. This model plays an important role in the theory of phase transitions and critical phenomena in physics, and has applications as widely varied as tumor migration, foam behaviors, and social demographics.

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V -polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including computational complexity results, to bear on a wider range of statistical mechanics models. 

This is joint work with Iain Moffatt.