Explosive connectivity and mechanical rigidity in cubic lattice structures

TL;DR

This work addresses how local edge-selection rules under a competitive Achlioptas product-rule process shape global connectivity and rigidity in 3D cubic lattices. By combining large-scale simulations on two distinct host geometries with a rigorous theoretical framework, the authors demonstrate a first-order explosive connectivity transition for $k\ge 2$ and reveal a host-dependent rigidity gap that can shrink with increasing $k$ in richly connected Intra networks. The theory introduces sublinear merger-cascade windows, a conditional progress function for rigidity, and synchronous coupling arguments that yield monotone delays in connectivity and monotone improvements in rigidification, including convergence to a maximally suppressed transition as $k\to\infty$. These results connect local competition rules to global mechanical stability, offering insights for designing rigid, architected materials and understanding network formation in jamming and biological contexts, with clear finite-size scaling signatures and practical implications for metamaterial fabrication.

Abstract

We study explosive connectivity and mechanical rigidity in three-dimensional cubic lattice structures under Achlioptas-type product-rule dynamics. Our work combines extensive numerical simulation with the development of a new theoretical framework. For connectivity, we rigorously establish the presence of sublinear-width merger-cascade windows for $k\ge 2$, which drive macroscopic jumps in the order parameter and imply a first-order transition. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ monotonically enhances the efficiency of rigidification. To explain this phenomenon, we propose a theoretical model centered on a conditional progress function that links an edge's local product-rule score to its global mechanical utility. We show that this function becomes non-increasing, thus explaining the observed monotonic efficiency, under two physically-motivated assumptions. Altogether, our work provides new insights into the relationship between local dynamics and global connectivity and rigidity in cubic lattice structures via both theory and computation.

Figures (6)

• Figure 1: The two 3D vertex models considered in this study. (a) The Nearest-Neighbor (NN) model. (b) The Intra-cube (Intra) model.
• Figure 2: An illustration of the k-choice Achlioptas process on an intermediate state of a cubic lattice structure with $L=2$ and $k=4$. (a) The Nearest-Neighbor (NN) model. (b) The Intra-cube (Intra) model. The existing edges from the previous states are in black. At the current state, four candidate edges (in grey and red) are sampled, and the one that minimizes the product score is selected (red).
• Figure 3: Order parameter transition for connectivity for varying choice k. (a)--(d) The NN model with $k = 1, 2, 8, 32$. (e)--(h) The Intra model with $k=1,2,8,32$. Each colored curve represents the average result of the 1,000 independent simulations for a specific system size $N=(L+1)^3$ from $L=1$ to $L=10$. For both NN and Intra models, increasing $k$ from 1 to 32 drives the system from a continuous-like transition of the order parameter $S_{\max}/N$ to a sharp, discontinuous jump, validating the theoretical prediction of a first-order transition for $k \ge 2$. See also SI Video S1--S2 for the results of all $k = 1, 2, \dots, 32$.
• Figure 4: Finite-size scaling of peak susceptibility. (a)--(d) The log-log plot of $\chi'_{\max}$ versus system size $N=(L+1)^3$ for the NN model with $k = 1$ (slope $\gamma = 0.900$), $k = 2$ ($\gamma = 1.182$), $k=8$ ($\gamma = 1.118$), $k = 32$ ($\gamma = 1.012$). (e)--(h) The log-log plot of $\chi_{\max}$ versus system size $N$ for the Intra model with $k = 1$ (slope $\gamma = 0.825$), $k = 2$ ($\gamma = 1.155$), $k=8$ ($\gamma = 1.118$), $k = 32$ ($\gamma = 0.978$). Each dot in each plot represents the average result of the 1,000 independent simulations. Note that the slope $\gamma$ transitions from a value characteristic of a second-order transition at $k=1$ to values near 1 for $k \ge 2$, indicating a crossover to a first-order regime. See also SI Video S3--S4 for the results of all $k = 1, 2, \dots, 32$.
• Figure 5: Exclusive susceptibility versus link density. (a)--(d) The NN model with $k = 1, 2, 8, 32$. (e)--(h) The Intra model with $k=1,2,8,32$. Each colored curve represents the average result of the 1,000 independent simulations for a specific system size $N=(L+1)^3$ from $L=1$ to $L=10$. The plots of the exclusive susceptibility $\chi'_L$ show that as $k$ increases, the peak shifts to higher densities, sharpens, and increases in height, corroborating the transition to first-order behavior. We plot $\chi'_L$ as its peak provides a clear numerical signature of the critical point $p_c$. See also SI Video S5--S6 for the results of all $k = 1, 2, \dots, 32$.

Theorems & Definitions (176)

• Definition 2.1: NN unit cell graph
• Definition 2.2: Intra unit cell graph
• Definition 2.3: Product score
• Definition 2.4: Product-rule with $k$ choices
• Definition 2.5: Rigidity matrix
• Definition 2.6: Trivial motions and floppy modes
• Definition 2.7: Generic placements and generic rigidity
• Definition 2.9: Connected components and sizes
• Definition 2.10: Susceptibility Measures
• Definition 2.11: Rank Gain and Edge Redundancy