On the study of jamming percolation

TL;DR

This paper analyzes kinetically constrained models of glassy transitions, identifying key model characteristics needed for rigorous proofs of discontinuous transitions and exploring their limitations in higher dimensions.

Contribution

It demonstrates the necessity of the 'No Parallel Crossing' rule for rigorous proofs and evaluates the applicability of the TBF proof to various jamming percolation models.

Findings

'No Parallel Crossing' rule is crucial for rigorous proof validity.

Most knight-like models fail the 'No Perpendicular Crossing' requirement.

The TBF proof does not extend straightforwardly to three-dimensional models.

Abstract

We investigate kinetically constrained models of glassy transitions, and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales. The models we investigate have constraints similar to that of the knights model, introduced by Toninelli, Biroli, and Fisher (TBF), but differing neighbor relations. We find that such knights-like models, otherwise known as models of jamming percolation, need a ``No Parallel Crossing'' rule for the TBF proof of a glassy transition to be valid. Furthermore, most knight-like models fail a ``No Perpendicular Crossing'' requirement, and thus need modification to be made rigorous. We also show how the ``No Parallel Crossing'' requirement can be used to evaluate the provable glassiness of other correlated percolation models, by looking at models with more stable directions than the knights model. Finally, we show that the TBF proof does not generalize in any straightforward fashion for three-dimensional versions of the knights-like models.