Directed rigidity and bootstrap percolation in (1+1) dimensions

TL;DR

This paper investigates directed rigidity percolation in (1+1) dimensions across different lattices, revealing a first-order transition at p=1 for square and triangular lattices, and a continuous transition with directed percolation universality on the augmented triangular lattice.

Contribution

It provides the first detailed analysis of directed rigidity percolation on multiple lattices, identifying universality classes and critical behaviors through simulations and analytical calculations.

Findings

Square and triangular lattices show first-order transition at p=1.

Augmented triangular lattice exhibits continuous transition in directed percolation class.

Analytical finite-size behavior near the discontinuous transition at p=1.

Abstract

We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a non-trivial p_c. On the augmented triangular lattice we find, by extensive numerical simulation, that the directed rigidity percolation transition belongs to the same universality class as directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e. the spreading of rigidity from finite clusters close to a non-rigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.