Is the Percolation Probability on $\mathbb{Z}^d$ with Long Range Connections Monotone?

TL;DR

This study numerically investigates how the percolation threshold in a long-range bond percolation model on a square lattice behaves as the range of connections increases, revealing monotonicity in two-range cases but complex behavior in three-range cases.

Contribution

First numerical estimations of the percolation threshold for two-range and three-range models, testing theoretical predictions and revealing unexpected non-monotonic behavior.

Findings

Percolation threshold is non-decreasing in two-range models and converges to the predicted value.

Three-range models show non-monotonic behavior for specific range combinations.

Results raise questions about the general monotonicity conjecture in long-range percolation.

Abstract

We present a numerical study for the threshold percolation probability, $p_c$, in the bond percolation model with multiple ranges, in the square lattice. A recent Theorem demonstrated by de Lima {\it et al.} [B. N. B. de Lima, R. P. Sanchis, R. W. C. Silva, STOCHASTIC PROC APPL {\bf 121}, 2043-2048 (2011)] states that the limit value of $p_c$ when the long ranges go to infinity converges to the bond percolation threshold in the hypercubic lattice, $\mathbb{Z}^d$, for some appropriate dimension $d$. We present the first numerical estimations for the percolation threshold considering two-range and three-range versions of the model. Applying a finite size analysis to the simulation data, we sketch the dependence of $p_c$ in function of the range of the largest bond. We shown that, for the two-range model, the percolation threshold is a non decreasing function, as conjectured in the cited work, and converges to the predicted value. However, the results to the three-range case exhibit a surprising non-monotonic behavior for specific combinations of the long range lengths, and the convergence to the predicted value is less evident, raising new questionings on this fascinating problem.