Lévy Flights: Faster Than Random, Sharper Than Chaos
Lévy Flights redefine how randomness operates in nature and computation—far exceeding Gaussian random walks with long, rare jumps that enable efficient exploration. Unlike classical Brownian motion, whose step lengths follow a normal distribution with finite variance, Lévy Flights exhibit heavy-tailed step-length distributions, leading to anomalous diffusion where mean squared displacement grows nonlinearly with time. This unique statistical structure makes them pivotal in modeling real-world systems such as animal foraging, neural search, and chaotic dynamics.
The Percolation Threshold and Critical Probability p_c ≈ 0.59274621Percolation theory on 2D square lattices reveals a profound phase transition at critical probability p_c ≈ 0.59274621—exactly the threshold where local connectivity fractures into global network connectivity.