Aftermath epidemics: Percolation on the sites visited by generalized random walks

Area of Science:

• Statistical Physics
• Complex Systems
• Probability Theory

Background:

• Percolation theory studies connectivity in random systems.
• Generalized random walks, including Levy flights and Knight's move walks, exhibit complex visitation patterns unlike ordinary random walks.
• The connectivity of visited sites in these walks presents a non-trivial percolation problem.

Purpose of the Study:

• To investigate percolation phenomena on finite lattices traversed by generalized random walks.
• To analyze the impact of different walk types (Levy flights, Knight's move) on percolation transitions.
• To determine critical exponents and universality classes for these non-trivial percolation systems.

Main Methods:

• Simulating generalized random walks (Levy flights, 2D/3D Knight's move walks) on finite lattices with periodic boundary conditions.
• Analyzing the connectivity of visited sites using percolation theory.
• Varying the density of visited sites (or number of steps) as a control parameter to identify percolation transitions.

Main Results:

• A continuous percolation transition was observed in most cases.
• Exceptions include 2D Knight's move random walks and Levy flights with Levy parameter σ≥2.
• 3D generalized Knight's move random walks fall into the pacman percolation universality class with apparent rational critical exponents (e.g., β=1).
• For 2D Levy flights (0<σ<2), scale invariance is broken, leading to significant finite-size scaling corrections and ambiguous critical exponent determination.

Conclusions:

• Generalized random walks create complex site visitation patterns that lead to non-trivial percolation.
• The universality class and existence of a true percolation transition depend on the specific type of generalized random walk and dimensionality.
• Further research is needed to resolve critical exponents in systems with broken scale invariance.