Flux-conserving directed percolation

TL;DR

This paper introduces a flux-conserving directed percolation model where the flux along bonds is dynamic, revealing power-law flux distributions and solvable cases, advancing understanding of directed percolation phenomena.

Contribution

The paper presents a novel flux-conserving directed percolation model with exact solutions in one dimension and approximations in two dimensions, linking flux dynamics to traditional percolation models.

Findings

Flux distribution follows a power law at small fluxes.

Exact solvability in one dimension due to independent site occupations.

Good approximation of flux behavior in two dimensions.

Abstract

We discuss a model for directed percolation in which the flux of material along each bond is a dynamical variable. The model includes a physically significant limiting case where the total flux of material is conserved. We show that the distribution of fluxes is asymptotic to a power law at small fluxes. We give an implicit equation for the exponent, in terms of probabilities characterising site occupations. In one dimension the site occupations are exactly independent, and the model is exactly solvable. In two dimensions, the independent-occupation assumption gives a good approximation. We explore the relationship between this model and traditional models for directed percolation.