High-precision Dynamic Monte Carlo Study of Rigidity Percolation

TL;DR

This paper introduces a dynamic pebble game algorithm to study the evolution of rigidity percolation on a lattice, revealing temporal self-similarity, cascade events, and providing high-precision critical parameters.

Contribution

It develops a computationally efficient dynamic approach to analyze rigidity percolation, uncovering temporal self-similarity and precise critical point estimates.

Findings

Identification of large-scale cascade events during bond addition

High-precision estimates of critical point and exponents

Discovery of temporal self-similarity in cluster evolution

Abstract

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 $1/\nu = 0.850(3)$, and the fractal dimension $d_f = 1.850(2)$, representing substantial improvements over existing values.