High-precision Dynamic Monte Carlo Study of Rigidity Percolation

Rigidity percolation provides an important basis for understanding the onset of mechanical stability in disordered materials. While most studies on the triangular lattice have focused on static properties at fixed bond~(site) occupation probabilities, the dynamics of the rigidity transition remain less explored. In this work, we formulate a dynamic pebble game algorithm that monitors how rigid clusters emerge and evolve as bonds are added sequentially to an empty lattice, with computational efficiency comparable to the standard static pebble game. We uncover a previously overlooked temporal self-similarity exhibited in multiple quantities, including the cluster size changes and merged cluster sizes during bond addition, as well as the number of simultaneously merging clusters. We identify large-scale cascade events in which a single bond addition triggers the merger of an extensive number of clusters that scales with system size with inverse correlation-length exponent. Using an event-based ensemble approach, we obtain high-precision estimates of the critical point $p_c = 0.660,277,8(10)$, the inverse correlation-length exponent $1/ν= 0.850(3)$, and the fractal dimension $d_f = 1.850(2)$, representing substantial improvements over existing values.

💡 Research Summary

This paper introduces a high‑precision dynamic Monte Carlo study of rigidity percolation (RP) on the two‑dimensional triangular lattice. While static investigations have long focused on fixed bond‑occupation probabilities, the authors recognize that the dynamics of the rigidity transition—how rigid clusters emerge, grow, and merge as bonds are added—remain largely unexplored. To address this gap, they develop a dynamic version of the pebble‑game (PG) algorithm that can track the full rigid‑cluster structure after each individual bond addition with computational efficiency comparable to the static PG.

The core of the algorithm consists of two innovations. First, persistent bidirectional data structures store the membership of each vertex in its current rigid cluster, allowing the independence of a newly added bond to be decided in O(1) time without repeated pebble searches. Second, during the identification of clusters that must merge, the authors employ a “let go of the largest” optimization: the largest cluster among those to be merged is excluded from the traversal, dramatically reducing the number of site visits required near criticality and in the super‑critical regime. Together these techniques bring the overall computational cost down to O(V^1.15), matching the performance of static PG implementations and enabling simulations up to linear size L = 8192 with periodic boundaries.

In the dynamic simulation, time is defined as t = T/E, where T is the number of bonds added and E the total number of bonds in the lattice. At each step the authors record (i) the change in cluster size (the “gap”), (ii) the number of clusters that merge as a result of the added bond, and (iii) the size of the largest rigid cluster C₁. Analysis of these time‑resolved observables reveals a previously unnoticed temporal self‑similarity: the distributions of gaps and of merged‑cluster counts follow universal power laws over many decades. Moreover, they identify large‑scale cascade events in which a single bond triggers the simultaneous merger of many clusters, including those not directly adjacent to the new bond—a hallmark of the non‑local nature of rigidity constraints. The maximum number of clusters merged in a single event, M_max, scales with system size as M_max ∼ L^{1/ν}, where 1/ν is the inverse correlation‑length exponent. This scaling provides a quantitative explanation for the unusually strong finite‑size corrections that have plagued previous RP studies.

To extract critical properties with maximal precision, the authors construct event‑based ensembles. For each realization they define a pseudo‑critical point t_L as the time at which a specific dynamical event occurs (e.g., the first appearance of a system‑spanning rigid cluster). By averaging over many realizations conditioned on such events, they obtain clean finite‑size scaling (FSS) behavior: t_L − t_c ∼ L^{−1/ν} and the fluctuations σ_{t_L} ∼ L^{−1/ν}. The same approach applied to C₁ yields consistent scaling. Using these ensembles, they determine the critical bond‑occupation probability p_c = 0.6602778(10), the inverse correlation‑length exponent 1/ν = 0.850(3), and the fractal dimension of the critical rigid cluster d_f = 1.850(2). These values improve upon earlier estimates by three orders of magnitude for p_c and significantly tighten the uncertainties for ν and d_f.

The paper concludes by emphasizing the broader implications of the dynamic PG algorithm and event‑based ensemble methodology. The algorithm makes large‑scale dynamic RP simulations feasible, opening the door to studies of other non‑local percolation problems, such as explosive percolation, high‑dimensional percolation, and rigidity transitions in realistic disordered materials (e.g., colloidal gels or biological tissues). The identified temporal self‑similarity and cascade scaling deepen our understanding of how non‑local constraints generate strong finite‑size effects, and the high‑precision critical parameters firmly place RP in a distinct universality class from ordinary Bernoulli percolation. Future work may extend the approach to three‑dimensional lattices, to networks with heterogeneous bond strengths, or to experimental systems where bond formation can be monitored in real time.