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Updated: Jun 25, 2026
08:01
The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
Field theory of directed percolation with long-range spreading
Hans-Karl Janssen1, Olaf Stenull
1Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany.
Summary
This study explores long-range interactions in spreading models using Lévy flights. It confirms four renormalization group fixed points and identifies stability lines, crucial for understanding phase transitions.
Area of Science:
- Statistical Physics
- Complex Systems
- Dynamical Processes
Background:
- Spreading models with short-range interactions typically exhibit phase transitions within the directed percolation (DP) universality class.
- Realistic spreading processes often involve long-range interactions, characterized by Lévy flights with a specific distance decay.
Purpose of the Study:
- To investigate directed percolation (DP) with long-range Lévy-flight spreading using advanced theoretical methods.
- To analyze the impact of long-range interactions on critical phenomena and universality classes in spreading models.
Main Methods:
- Application of renormalized field theory techniques.
- Analysis of renormalization group fixed points and their stability regions in the (sigma,d) plane.
- Calculation of critical exponents and leading logarithmic corrections using an epsilon expansion.
Main Results:
- Confirmation of four distinct renormalization group fixed points: short-range Gaussian, Lévy Gaussian, short-range DP, and Lévy DP.
- Identification of four lines in the (sigma,d) plane delineating the stability regions of these fixed points.
- Demonstration of continuous changes in critical exponents when crossing the stability line between short-range and Lévy DP.
Conclusions:
- The study provides a robust theoretical framework for understanding spreading phenomena with long-range interactions.
- Results corroborate and extend previous findings on Lévy-flight-driven directed percolation.
- Calculated critical exponents and corrections offer valuable benchmarks for numerical simulations and future research.
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