On the time complexity analysis of numerical percolation threshold estimation

The main purpose of percolation theory is to model phase transitions in a variety of random systems, which is highly valuable in fields related to materials physics, biology, or otherwise unrelated areas like oil extraction or even quantum computing. Thus, one of the problems encountered is the calculation of the threshold at which such transition occurs, known as percolation threshold. Since there are no known closed forms to determine the threshold in an exact manner in systems with particular properties, it is decided to rely on numerical methods as the Monte Carlo approach, which provides a sufficiently accurate approximation to serve as a valid estimate in the projects or research where it is involved. However, in order to achieve an exact characterization of the threshold in two-dimensional systems with site percolation, in this work it is performed an analysis of the complexity, both temporal and spatial, of an algorithm that implements its computation from the aforementioned numerical method. Specifically, the conduction of an accurate analysis of the cost of such algorithm implies a deep enough knowledge about certain metrics regarding its duration, or work completed per iteration, which along with its formalization may contribute to the determination of the threshold based on these metrics. Nevertheless, as a result, various bounds are achieved for the best, average and worst cases of the execution on systems spanning several dimensions, revealing that in 1 and 2 the complexity is directly conditioned by the duration, although from 3 onwards no proof for this point has been found, notwithstanding the evidence suggesting its compliance. Furthermore, based on the average case, several methods are proposed that could be applied to characterize the threshold, although they have not been thoroughly explored beyond what is necessary for the complexity analysis.