On the time complexity analysis of numerical percolation threshold estimation
Daniel García Solla
TL;DR
This work analyzes the temporal and spatial complexity of a Monte Carlo algorithm for estimating the site-percolation threshold on $n\times n$ grids with Moore neighborhoods. It develops a framework that separates per-iteration costs (insertion-based DFS traversals) from the number of iterations needed to form a spanning cluster, and proposes practical estimators for average cluster size $c(n,k)$ via continuous and discrete scalar-field approaches. The authors derive exact and bound-oriented results in 1D, and extend (with substantial challenges) to 2D, including empirical time measurements that validate best- and worst-case bounds and motivate average-case modeling. Key contributions include explicit bounds like $T_{best}(n)=O(n\log n)$ in 1D, $T_{worst}(n)=\Theta(n^2)$ in 1D, and $T_{worst}(n)=\Theta(n^4)$ in 2D, alongside multiple estimators for the average work per insertion and the associated probabilistic characterizations of termination. These results offer guidance for resource planning and algorithm design in percolation threshold estimation and point to fruitful directions for more accurate 2D modeling and potential closed-form characterizations of terminal-state distributions.
Abstract
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.