Combinatorial models of rigidity and renormalization

TL;DR

This paper introduces a renormalization approach to solve $(k,l)$-percolation problems on hierarchical graphs, connecting graph theory with physical concepts of percolation and revealing phase transitions in rigidity and ordinary percolation.

Contribution

It develops a renormalization transformation for $(k,l)$-percolation problems and demonstrates its exact solution on hierarchical graphs for certain parameter ranges.

Findings

Exact solutions for $(k,l)$-percolation on hierarchical graphs.

Identification of phase transitions between ordinary and rigidity percolation.

The renormalization method's domain of validity is established.

Abstract

We first introduce the percolation problems associated with the graph theoretical concepts of $(k,l)$-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for $(k,l)$-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of $(k,l)$-percolation problems on hierarchical graphs, for $k\leq l<2k$. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values of the parameters.