Scaling relations and finite-size scaling in gravitationally correlated lattice percolation models

Abstract

In some systems, the connecting probability (and thus the percolation process) between two sites depends on the geometric distance between them. To understand such process, we propose gravitationally correlated percolation models for link-adding networks on the two-dimensional lattice G with two strategies S max and S min , to add a link l i,j to connect site i and site j with mass m i and m j , respectively; m i and m j are sizes of the clusters which contain site i and site j, respectively.The probability to add the link l i,j is related to the generalized gravity g ij ≡ m i m j /r d ij , where r ij is the geometric distance between i and j, and d is an adjustable decaying exponent. In the beginning of the simulation, all sites of G are occupied and there is no link. In the simulation process, two inter-cluster links l i,j and l k,n are randomly chosen and the generalized gravities g ij and g kn are computed. In the strategy S max , the link with larger generalized gravity is added. In the strategy S min , the link with smaller generalized gravity is added, which include percolation on the Erdős-Rényi random graph and the Achlioptas process of explosive percolation as the limiting cases, d → ∞ and d → 0, respectively. Adjustable strategies facilitate or inhibit the network percolation in a generic view. We calculate percolation thresholds T c and critical exponents β by numerical simulations. We also obtain various finite-size scaling functions for the node fractions in percolating clusters or arrival of saturation length with different intervening strategies.

Relevant alternative analytic average magnetization calculation method for the square and the honeycomb Ising lattices

“…The Ising model, which was originated by Lenz in 1920 and was subsequently investigated by his student Ising in 1925 [1], unifies the study of phase transitions in systems as divers as ferromagnet, gas-liquids, binary alloys, and so on [2,3,4,5,6]. In one-dimension (1D), it has exact solution with the prediction of the absence of phase transition at finite temperature [7,8].…”

Analytic average magnetization expression for the body centered cubic Ising lattice

“…Later, it was investigated by his student Ising in 1925 [1]. The model is very important in the investigation of phase transition since it unifies the study of phase transitions in systems as diverse as gas-liquids, ferromagnets, binary alloys, and so on [2][3][4][5][6]. The model has an exact solution in one-dimension (1D) with the prediction of the absence of phase transition at finite temperature [7][8][9][10][11].…”