On $k$-Core Percolation in Four Dimensions

TL;DR

This paper investigates the nature of $k$-core percolation transitions in four dimensions, revealing a first-order transition for $k=4$ and a continuous transition for $k=3$, with detailed numerical analysis.

Contribution

It provides the first detailed numerical study of $k$-core percolation on a four-dimensional lattice, clarifying the order of phase transitions for different $k$ values.

Findings

$k=4$ transition is strictly first order with no singular behavior before the jump.

$k=3$ transition is continuous.

Correlation length remains finite at the $k=4$ transition.

Abstract

The $k$-core percolation on the Bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first-order/second-order nature. We investigate numerically $k$-core percolation on the four-dimensional regular lattice. For $k=4$ the presence of a discontinuous transition is clearly established but its nature is strictly first order. In particular, the $k$-core density displays no singular behavior before the jump and its correlation length remains finite. For $k=3$ the transition is continuous.