Interpretation of percolation in terms of infinity computations

TL;DR

This paper introduces a novel computational approach using the Infinity Computer to analyze percolation models, revealing that phase transitions are intervals rather than points and multiple infinite clusters can coexist near the threshold.

Contribution

It presents a new methodology for studying percolation using infinite and infinitesimal numbers, expanding traditional models without contradicting Cantor's ideas.

Findings

Phase transition point becomes a critical interval in the new approach

Multiple infinite clusters coexist near the percolation threshold

The new method allows numerical work with infinite and infinitesimal quantities

Abstract

In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor's ideas and describes infinite and infinitesimal numbers in accordance with the principle `The part is less than the whole'. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer - the Infinity Computer - introduced recently in by Ya.D. Sergeyev in a number of patents. The new approach does not contradict Cantor. In contrast, it can be viewed as an evolution of his deep ideas regarding the existence of different infinite numbers in a more applied way. Site percolation and gradient percolation have been studied by applying the new computational tools. It has been established that in an infinite system the phase transition point is not really a point as with respect of traditional approach. In light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on "microscope" we use this interval could be regarded as finite, infinite and infinitesimal short interval. Using new approach we observed that in vicinity of percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional consideration.