The Geometry of Fixed-Magnetization Spin Systems at Low Temperature

TL;DR

The paper studies the geometry of fixed-magnetization spin configurations for the Generalized Potts Model on a triangular lattice at low temperature, showing that typical configurations decompose into compact, nearly monochromatic regions with nearly minimal boundaries. It develops a novel four-step strategy to relate fixed-magnetization analysis to the better-understood variable-magnetization setting via a balancing magnetic field, truncated contour weights, and a special contour set derived from optimal density perturbations. The main contribution is a rigorous proof that, with high probability, configurations under π_{G,β,ρ} are Sorted into regions with short boundaries and near-target color densities, extending Peierls and Pirogov–Sinai methods to more general, fixed-magnetization spin systems and to applications in programmable matter and cellular Potts models. The work provides new techniques for comparing fixed- and variable-magnetization partition functions, enabling low-temperature geometric estimates beyond the classic Ising/Potts cases, and lays groundwork for algorithmic and modeling advances in programmable matter and computational biology.

Abstract

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.

Figures (3)

• Figure 1: Configurations of various $8$-color instances of the F-GPM on a $63 \times 63$ domain of the triangular lattice with periodic boundary conditions, $\beta = 1$, and density vector ${\boldsymbol{\rho}} \propto (1,1,\dots,1,14)$. The first example is the Potts model, while the last is the clock model.
• Figure 2: Configurations of various $9$-color instances of the F-GPM on a $63 \times 63$ domain of the triangular lattice with periodic boundary conditions, $\beta = 1$, and density vector ${\boldsymbol{\rho}} \propto (1,1,\dots,1,16)$. The cost matrices are listed in \ref{['apx:interactionmatrices']}. (a) Pairs of the form $(i,i+1)$ for $i \in \{1,3,5,7\}$ greatly attract, while the groups of $1$--$4$ and $5$--$8$ moderately attract. (b) Colors $1$--$4$ and $5$--$8$ are sorted, but these groups do not attract. (c) Like (b), except the groups attract. (d) Colors $2$--$8$ are incentivized to sort into a loop, while color $1$ aims to avoid color $9$.
• Figure 3: A contour $\gamma = (\overline{\gamma}, \sigma_{\overline{\gamma}}) \in \mathcal{C}_0$, defined over the infinite lattice ${\Lambda^\infty}$. The contour divides the lattice into $\overline{\gamma}$, the exterior $\mathrm{Ext}(\gamma)$ (of type $1$) and the interiors $\mathrm{Int}_{i}(\gamma), i \in [q]$. The assignment $\sigma_{\overline{\gamma}}$ applies colors to each vertex in $\overline{\gamma}$, which are shown in red. These colors dictate the label of each component of the subgraph of ${\Lambda^\infty}$ induced by $V({\Lambda^\infty})\setminus \overline{\gamma}$.

Theorems & Definitions (54)

• Definition 1.1: The $\mathsf{Sorted}$ event
• Theorem 1: Main
• Corollary 1.2
• proof
• Definition 3.1: Truncated Pressure
• Definition 3.2: $\tau$-stable
• Lemma 3.3: Balancing Pressures and Stability
• proof
• Lemma 3.4: Equivalence of truncated and untruncated partition functions
• proof