Detection Percolation (discontinuous Lévy)

Detection Percolation

Poisson particle system in a torus of side \(L\). Stop when the target (at the origin) is covered. In this variant Lévy motion is a true jump process — particles rest, then leap instantly, with a fading line marking each jump — while Brownian motion stays continuous.

Box & intensity

Box side \(L\)300

Intensity \(\lambda\)2.0×10⁻³

\(\mathbb{E}[N] = \lambda L^2\). Currently \(\approx\) 180.

Motion

Brownian Lévy

Diffusion \(\sigma\)1.0

Gaussian increment per tick: \(\Delta x = \sigma\sqrt{\Delta t}\,Z\).

Stability \(\alpha\)1.50

Heavy-tail jumps with \(P(|J|>x)\sim x^{-\alpha}\), drawn as instantaneous leaps.

Radius

Fixed \(r\) Pareto(\(\delta\))

Radius \(r\)3.0

Target particle

Fixed at origin Moving from origin

Stopping scenario

1 — any particle 2 — largest component

Scenario 1. Stop as soon as the target lies in some disc \(B(x_i,r_i)\).

Playback

Speed (ticks / frame)3

Random seed