Critical probability of percolation over bounded region in N-dimensional Euclidean space

TL;DR

This paper introduces an analytical model based on excursion set theory to predict the critical probability of percolation in bounded regions across N-dimensional Euclidean spaces, with detailed analysis for 3D and generalization to higher dimensions.

Contribution

It extends existing models by incorporating N-dimensional bounded regions and provides a generalized analytical framework for percolation critical probability prediction.

Findings

Derived analytical expressions for critical percolation probability in 3D.

Calculated statistically representative volume elements for the 3D case.

Generalized the model to N-dimensional Euclidean spaces.

Abstract

Following H. Tomita and C. Murakami we propose an analytical model to predict critical probability of percolation. It is based on the excursion set theory which allows us to consider N-dimensional bounded regions. Details are given for the 3D case and statistically Representative Volume Elements are calculated. Finally generalisation to the N-dimensional case is made.