Number fluctuations in 2-D uncorrelated and correlated point patterns: a computational revision

Number fluctuations in two-dimensional point patterns are revisited through the variance of the particle number counted in square observation windows, Var(N) ∝ ⟨N⟩^α, with α = 1 for uncorrelated (Poisson) patterns, α < 1 for effectively repulsive correlations and α > 1 for effectively attractive ones, and with P(N) approaching a Gaussian as ⟨N⟩ grows. This revision of the original work rebuilds the simulation engine: a spatial-grid neighbour search and an integral-image window-counting scheme yield measured speedups of roughly 30–40× (of order 1000× for pattern generation relative to the original mask-based code), removing the loss of statistics at large ⟨N⟩. The added statistics refine the α(c) curve, revealing a sharp repulsive minimum at c* = 0.756 ± 0.004 and a broad attractive plateau rather than symmetric extrema at c = 1. The third and fourth moments of P(N) are shown to follow the Poisson laws quantitatively in the random case, and the model is placed within Torquato's hyperuniformity classification, where effective repulsion achieves only weak fluctuation suppression. This record contains the revised paper (2026), the original 2024 undergraduate thesis (Treball de Fi de Grau, Universitat de Barcelona) as an annex, the presentation slides, the Python simulation engine (pointfluct.py), and a MATLAB-vs-Python code comparison.