The Geometry of Fixed-Magnetization Spin Systems at Low Temperature

Spin systems are fundamental models of statistical physics that provide insight into collective behavior across scientific domains. Their interest to computer science stems in part from the deep connection between the phase transitions they exhibit and the computational complexity of sampling from the probability distributions they describe. Our focus is on the geometry of spin configurations, motivated by applications to programmable matter and computational biology. Rigorous results in this vein are scarce because the natural setting of these applications is the low-temperature, fixed-magnetization regime. Recent progress in this regime is largely limited to spin systems under which magnetization concentrates, which enables the analysis to be reduced to that of the simpler, variable-magnetization case. More complicated models, like those that arise in applications, do not share this property. We study the geometry of spin configurations on the triangular lattice under the Generalized Potts Model (GPM), which generalizes many fundamental models of statistical physics, including the Ising, Potts, clock, and Blume--Capel models. Moreover, it specializes to models used to program active matter to solve tasks like compression and separation, and it is closely related to the Cellular Potts Model, widely used in computational models of biological processes. Our main result shows that, under the fixed-magnetization GPM at low temperature, spins of different types are typically partitioned into regions of mostly one type, separated by boundaries that have nearly minimal perimeter. The proof uses techniques from Pirogov--Sinai theory to extend a classic Peierls argument for the fixed-magnetization Ising model, and introduces a new approach for comparing the partition functions of fixed- and variable-magnetization models.