Directed Percolation with a Wall or Edge

TL;DR

This paper studies how introducing a wall or edge affects directed percolation, connecting it to surface critical phenomena, and introduces new exponents with theoretical and numerical analysis.

Contribution

It presents scaling ansatzes for clusters near walls and edges in directed percolation, linking to surface critical phenomena and deriving new exponents.

Findings

Surface exponents satisfy generalized scaling relations.

Mean-field and numerical exponents for edge directed percolation are provided.

Numerical measurements confirm theoretical predictions.

Abstract

We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatzes are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalising those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2+1 dimensions.