The square sticky disk: crystallization and Gamma-convergence to the octagonal anisotropic perimeter

TL;DR

This work analyzes a square-norm variant of the sticky disk model, proving that energy minimizers crystallize on the square lattice for any fixed particle count and, in the large-N limit, exhibit a unique orientation and a Gamma-limit to an anisotropic perimeter. The authors develop a local, graph-based energy decomposition that extends the De Luca–Friesecke framework to the $\|\,\cdot\|_\infty$ metric, enabling sharp lower bounds and a clear link between combinatorial face defects and continuum anisotropy. The Gamma-limit is the anisotropic perimeter $P_\phi$ with $\phi(\nu)=|\nu_1|+|\nu_2|+|\nu_1+\nu_2|+|\nu_1-\nu_2|$, whose Wulff shape is the regular octagon, reflecting the square-norm geometry. Additionally, a compactness result shows that almost-minimizing sequences converge to a finite-perimeter set with a single orientation, underscoring a robust crystallization phenomenon in the anisotropic setting. Overall, the paper provides a rigorous bridge from discrete square-face interactions to a continuum variational limit with octagonal anisotropy, advancing understanding of crystallization under non-Euclidean metrics.

Abstract

We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm $\lVert\cdot\rVert_\infty$ in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number $N$ of particles. Then we consider the limit as $N\to\infty$: in contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the $Γ$-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [J. Nonlinear Sci. 28 (2018), 69-90] in the triangular case.

Figures (9)

• Figure 1: An admissible graph: the graph has straight edges and is planar, with the only possible exception of convex quadrilateral faces having both diagonals.
• Figure 2: Example of a possible face $F$ (shaded in grey). The angles that contribute to $\delta(F)$ are highlighted in red. In this case $\delta(F)=3\mathrm{P}_\mathrm{comb}(F)-8+8\#\mathcal{C}^{\mathrm{int}}( F)=3 \mathrm{P}_\mathrm{comb}(F)+8$, and $\mathrm{P}_\mathrm{comb}(F)=27$. Notice that in $\# \mathcal{C}^\mathrm{int}(F)$ we count the number of interior connected components of $\partial F$ (2 in this case), and not the number of interior connected components of $\mathbb{R}^2\setminus F$ (3 in this case).
• Figure 3: The two types of minimizers of the energy for $N=5$. On the left: the configuration $\{(0,0),(1,0),(0,1),(-1,0), (0,-1)\}$. On the right: the configuration $X_4\cup\{\bar{x}\}$, where $\bar{x}\in I:=\{(2,t)\,:\,t\in [0,1]\}$; the segment $I$ is represented by the dashed line.
• Figure 4: Reference for the proof of Proposition \ref{['basicprop']}. The wire edge is $\{x,x'\}$. By "sliding" the whole component $\mathsf{C}'$ along $S(x)$ (dotted) we can create a new bond (shown in red) thus proving that the original configuration is not minimal.
• Figure 5: Proof of Lemma \ref{['angoliconcavi']}.

Theorems & Definitions (43)

• Definition 1
• Lemma 2
• proof
• Theorem 3: Energy decomposition for 8 neighbors
• proof
• Remark 4
• Remark 5
• Theorem 6: Energy decomposition for 6 neighbors
• Remark 7: Relation with the energy decomposition in DeLucaFriesecke
• Theorem 8