Percolation Transition and Related Phenomena in Terms of Grossone Infinity Computations

Abstract

In this chapter, a number of traditional models related to the percolation theory is taken into consideration: site percolation, gradient percolation, and forest-fire model. They are studied by means of a new computational methodology that 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. It is established that in light of the new arithmetic using grossone-based numerals the phase transition point in site percolation and gradient percolation appears as a critical interval, rather than a critical point. Depending on the ‘microscope’ we use, this interval could be regarded as finite, infinite, or infinitesimal interval. By applying the new approach we show that in vicinity of the percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional considerations. With respect to the cellular automaton forest-fire model, two traditional versions of the model are studied: a real forest-fire model where fire catches adjacent trees in the forest in the step by step manner and a simplified version with instantaneous combustion. By applying the new approach there is observed that in both situations we deal with the same model but with different time resolutions. We show that depending on ‘microscope’ we use, the same cellular automaton forest-fire model reveals either the instantaneous forest combustion or the step by step firing.