Surface and bulk criticality in midpoint percolation

TL;DR

This study investigates the critical behavior of percolation at the midpoint in hypercubic lattices up to 10 dimensions, revealing how surface and bulk properties differ at and above the upper critical dimension.

Contribution

It introduces a boundary-inclusive approach to analyze critical indices in high-dimensional lattices, extending understanding of surface and bulk percolation phenomena.

Findings

Critical indices can be obtained via boundary analysis.

Percolation clusters have finite surface points above dimension 6.

Surface points approach 2d in large dimensions.

Abstract

The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular $d$-dimensional hypercubic lattices are in the present work investigated using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to $d=10$ by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension $d=6$, it is found that the percolation cluster contains only a finite number of surface points in the infinite-size limit. This is in accordance with the expectation from studies of lattices with negative curvature. It is also found that the number of surface points, reached by the percolation cluster in the infinite limit, approaches 2d for large dimensions $d$. We also note that the size dependence in proliferation of percolating clusters for $d\ge 7$ can be obtained by solely counting surface points of the midpoint cluster.