Hierarchical models of rigidity percolation

TL;DR

This paper introduces exactly solvable hierarchical models of 2D rigidity percolation, examining how self-organization affects phase diagrams, and finds no intermediate phase unlike in some previous studies.

Contribution

It presents a new exactly solvable hierarchical model for 2D rigidity percolation and explores the impact of self-organization on phase behavior.

Findings

No true intermediate phase found in the hierarchical models

Self-organization does not induce a phase separation in these models

Exact solutions provide insights contrasting with previous numerical studies

Abstract

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks, and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.