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Directed rigidity and bootstrap percolation in 1+1 dimensions.
M A de Menezes1, C F Moukarzel
1Instituto de Física, Universidade Federal Fluminense, CEP 24210-340, Niteroi, RJ, Brazil. marcio@if.uff.br
Summary
Directed rigidity percolation on different lattices reveals distinct phase transitions. The augmented triangular lattice shows a continuous transition belonging to the directed percolation universality class.
Area of Science:
- Statistical Physics
- Complex Systems
- Percolation Theory
Background:
- Directed rigidity percolation, also known as directed bootstrap percolation, is a model for the formation of rigid structures.
- Understanding phase transitions in such models is crucial for various fields, including materials science and network theory.
Purpose of the Study:
- To investigate the behavior of directed rigidity percolation on square, triangular, and augmented triangular lattices.
- To determine the universality class of the transition on the augmented triangular lattice.
- To analytically calculate finite-size effects near the transition on the triangular lattice.
Main Methods:
- Numerical simulations on different lattice structures.
- Analysis of surface critical behavior.
- Analytical calculations of finite-size scaling.
Main Results:
- Square and triangular lattices exhibit a first-order phase transition at p=1.
- The augmented triangular lattice displays a continuous phase transition at a nontrivial critical probability p(c).
- The transition on the augmented triangular lattice belongs to the same universality class as directed percolation.
- Finite-size behavior of rigid site density near the p=1 transition on the triangular lattice was analytically determined.
Conclusions:
- The augmented triangular lattice provides a new system for studying directed percolation universality.
- Analytical and numerical results confirm the distinct behaviors of directed rigidity percolation on different lattices.