Long-range k-core percolation

Bootstrap (or k-core) Percolation with k = 3 is studied numerically on two-dimensional lattices with long-range links whose lengths r_ij are distributed according to P(r) ~ r^−α. By varying the decay exponent α the topology of these networks can be made to range from two-dimensional short-range networks to ∞-dimensional random graphs. The 3-core transition is found to be of first-order character with a divergent correlation length for α < 2.75 and of second order for larger α. Whenever the transition is first-order an associated critical corona is found to exist. The correlation length exponent ν defined from the corona correlation length above the first order transition is estimated as v ≈ 1/2 for α = 0, and only shows a weak α dependence for α ≤ 2.50. The second-order transition at large α is found to be in the universality class of two-dimensional Percolation.