1∕dexpansion fork-core percolation

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 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 ≥ 3-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.

Complete set of types of phase transition in generalized heterogeneousk-core percolation

“…as the susceptibility [20,[22][23][24], which scales as χ (p) = lim N→∞ χ (p,N) ∼ (p − p * ) −γ in a continuous transition.…”

“…(20) to the simulation data of p max (N ) as shown in Fig. 4(a), p * andν 4 are estimated as p * 0.400(2) andν 2.4 (1).…”

“…Since HKC percolation considers only the maximal (giant) cluster, we use the fluctuation of the order parameter as the susceptibility. The fluctuation of the order parameter has widely been used to calculate γ in studies on the various percolation problems [20][21][22][23][24] as well as the mean size of finite clusters [25].…”

Constraint percolation on hyperbolic lattices

“…In other words, k = 3-core percolation may be very sensitive to loops. A 1/d expansion for k = 3core percolation demonstrated that the mixed nature of the transiton remained to order 1/d 3 [43]. Of course, the loops are controlled perturbatively in this 1/d expansion, which is not the case for the hyperbolic tessellation.…”

Fate of the Hybrid Transition of Bootstrap Percolation in Physical Dimension

“…Nevertheless these negative results leave the question open for the countless other lattices (either regular or random) in any dimension. Further-more a perturbative expansion around the large dimension limit of [19] suggested that both the MF transition and its hybrid character survives.…”