Generating Function for Particle-Number Probability Distribution in Directed Percolation

TL;DR

This paper derives a comprehensive generating function for particle-number distributions in directed percolation models, incorporating mean-field and renormalization group analyses, and confirms universality of the scaling form numerically.

Contribution

It provides a new analytical expression for the generating function of particle-number distributions in directed percolation, including corrections at the upper critical dimension.

Findings

Derived a generic generating function for PNPD in directed percolation.

Identified logarithmic corrections at the upper critical dimension.

Numerically confirmed the universality of the critical scaling form.

Abstract

We derive a generic expression for the generating function (GF) of the particle-number probability distribution (PNPD) for a simple reaction diffusion model that belongs to the directed percolation universality class. Starting with a single particle on a lattice, we show that the GF of the PNPD can be written as an infinite series of cumulants taken at zero momentum. This series can be summed up into a complete form at the level of a mean-field approximation. Using the renormalization group techniques, we determine logarithmic corrections for the GF at the upper critical dimension. We also find the critical scaling form for the PNPD and check its universality numerically in one dimension. The critical scaling function is found to be universal up to two non-universal metric factors.