$1/d$ Expansion for $k$-Core Percolation

TL;DR

This paper develops a $1/d$ expansion to analyze the nature of $k$-core percolation transitions in high-dimensional lattices, revealing that the hybrid transition persists beyond mean field theory.

Contribution

It introduces a $1/d$ expansion method for $k$-core percolation on hypercubic lattices, showing the hybrid transition survives in high dimensions.

Findings

Singularity in order parameter and susceptibility coincide at order $1/d^3$

Hybrid transition persists in high-dimensional systems

Results support mean field predictions in finite dimensions

Abstract

The physics of $k$-core percolation pertains to those systems whose constituents require a minimum number of $k$ connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para ${\rm H}_2$ mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary ($k=1$) and biconnected ($k=2$) percolation, the mean field $k\ge3$-core percolation transition is both continuous and discontinuous, i.e. there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a $1/d$ expansion for $k$-core percolation on the $d$-dimensional hypercubic lattice. We show that to order $1/d^3$ the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field $k$-core transition survives in high dimensions.