Occupied Neighbors Research Articles

Critical exponents for high density and bootstrap percolation

In high density percolation, clusters consisting only of occupied sites with a given minimum number m of occupied neighbours are studied. In bootstrap percolation, sites are first occupied at random, and then all occupied sites with less than m occupied neighbours are successively rendered unoccupied, until a stable configuration is reached. Both problems are studied on the square and triangular lattice, for all possible values of m, with the use of position-space renormalisation-group technique. Approximate estimates are found for the percolation threshold pc and the critical exponents nu and beta in all cases, which support universal behaviour for the high density problem as a function of m, and non-universal behaviour for the bootstrap problem. Comments are made on the application of different techniques for the calculation of beta on the onset of first-order transitions within the context of a position space-renormalisation-group approach.

Critical slowing down in dynamic bootstrap percolation

In bootstrap percolation the sites on a lattice are first occupied with probability p, then the occupied sites with less than a specified number m of occupied neighbours are successively culled until a stable configuration is reached. For m=4 on a cubic lattice, a previous computer study has yielded a first-order transition, in agreement with an exact solution on a Cayley tree; however, this study could not unambiguously detect the critical fluctuations present in the Cayley-tree solution. The authors present a phenomenological theory of the time dependence of the percolation probability as neighbours are culled. This mean-field theory is constructed to be consistent with the Cayley-tree results in the static limit and predicts a 'critical slowing down' of the order parameter as p to pc+ with r varies as (p-pc)-x and x=1/2. The authors present computer data for m= infinity on a simple cubic lattice which shows such critical slowing down, but with x=2.3+or-0.7. This suggests that on real lattices critical fluctuations in the bootstrap transition may be superimposed on the order-parameter discontinuity, as predicted by the Cayley-tree solution.

High-density site percolation on real lattices

In high-density site percolation, attention is centred only on those clusters consisting of occupied sites with a given minimum number m of occupied neighbours. The authors study this problem on square, triangular and cubic lattices using Monte Carlo computer simulations and generate several new results. High-density site percolation thresholds are given to moderate accuracy for all relevant values of m on the three lattices considered. They also find their results consistent with universality as the integer m is varied. Specifically, the critical exponents beta and gamma are shown to be determined mainly by the dimensionality d of the lattice in question and insensitive to variations in the value of the parameter m.

Bootstrap percolation transitions on real lattices

In bootstrap percolation, a fraction p of the sites of a lattice is first randomly occupied, and then all occupied sites with fewer than a given integral number m of occupied neighbours are successively culled (rendered unoccupied) until either (a) a configuration of occupied sites, each with m or more occupied neighbours, ensues or (b) all sites of the lattice are vacated. The authors treat this problem as a function of m on square, triangular and cubic lattices by computer simulation. Various types of percolation transitions are discovered, including some which are discontinuous and some which are continuous but characterised by non-universal critical exponents.

Bootstrap percolation on a Bethe lattice

A new percolation problem is posed which can exhibit a first-order transition. In bootstrap percolation, sites on an empty lattice are first randomly occupied, and then all occupied sites with less than a given number m of occupied neighbours are successively removed until a stable configuration is reached. On any lattice for sufficiently large m, the ensuing clusters can only be infinite. On a Bethe lattice for m>or=3, the fraction of the lattice occupied by infinite clusters discontinuously jumps from zero at the percolation threshold. From an analysis of stable and metastable ground states of the dilute Blume-Capel model (1966), it is concluded that effects like bootstrap percolation may occur in some real magnets.

High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponentsβ=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp c (m=1)=l/(z − 1) andp c(m=z) =[l/(z − 1)]1/(z−1). The cluster size distribution asymptotically decays exponentially withn, for largen, p ≠ p c .