Percolation Threshold Estimation Via Probabilistic Bounds And Simulation
Authors: Hanumesha S T
Abstract: Percolation theory provides a mathematically elegant and practically powerful framework for modeling connectivity transitions in random media, with applications ranging from porous materials and composite conductivity to epidemics, network robustness, and transport in disordered systems. A central quantity is the percolation threshold p_c, the critical occupation probability at which macroscopic connectivity emerges with nontrivial scaling. Although p_c is known exactly for a few planar cases and lattices, many practical scenarios require estimation under finite-size, boundary, and uncertainty constraints. This paper develops a rigorous and computation-oriented methodology for percolation threshold estimation that couples (i) probabilistic inequalities and bracketing arguments (crossing probabilities, monotonicity, sharp-threshold heuristics, and finite-size scaling), with (ii) simulation-based estimators (spanning probability curves, union-find connectivity, confidence intervals, and extrapolation). We emphasize a "two-engine" approach: bounds that constrain plausible threshold locations and simulation that refines the estimate while quantifying uncertainty. We also introduce an uncertainty-aware parameterization using intuitionistic fuzzy sets and (hyper)graph abstractions to represent ambiguous occupancy mechanisms and heterogeneous coupling patterns; this is motivated by real settings where the effective "open probability" is not a crisp scalar but a range informed by measurement noise or multi-factor criteria. The final manuscript provides a Word-ready, mathematics-forward exposition, with figures and tables embedded to illustrate lattice configurations, spanning curves, scaling collapse, and probabilistic bracketing.
DOI: http://doi.org/10.5281/zenodo.18092138
